Quantum query complexity of symmetric oracle
problems.
Daniel Copeland
1
and Jamie Pommersheim
2
1
UC San Diego
2
Reed College
We study the query complexity of quantum learning problems in which the oracles
form a group G of unitary matrices. In the simplest case, one wishes to identify the
oracle, and we find a description of the optimal success probability of a t-query quantum
algorithm in terms of group characters. As an application, we show that Ω(n) queries
are required to identify a random permutation in S
n
. More generally, suppose H is
a fixed subgroup of the group G of oracles, and given access to an oracle sampled
uniformly from G, we want to learn which coset of H the oracle belongs to. We call
this problem coset identification and it generalizes a number of well-known quantum
algorithms including the Bernstein-Vazirani problem, the van Dam problem and finite
field polynomial interpolation. We provide character-theoretic formulas for the optimal
success probability achieved by a t-query algorithm for this problem. One application
involves the Heisenberg group and provides a family of problems depending on n which
require n + 1 queries classically and only 1 query quantumly.
1 Introduction
An oracle problem is a learning task in which a learner tries to determine some information by
asking certain questions to a teacher, called an oracle. In our setting the learner is a quantum
computer and the oracle is an unknown unitary operator acting on some subsystem of the com-
puter. The computer asks questions by preparing states, subjecting them to the oracle, measuring
the results, and finally making a guess about the hidden information. How many queries to the
oracle are needed by the computer to guess the correct answer with high probability?
This paper addresses the following oracle problem. Fix a finite group G and a subgroup H ≤ G.
The elements of G are encoded as unitary operators by some unitary representation π : G → U(V ).
Given oracle access to π(a) (for some unknown a ∈ G) the learner must guess which coset of H
the element a lies in. We focus on average case success probability, though an easy averaging ar-
gument, given in Section 2, shows that the worst case and average case query complexity are equal.
We call this problem coset identification. This task encompasses many of previously studied
qauntum oracle problems, including univariate and multivariate polynomial interpolation over a
finite field [CvDHS16, CCH18], the group summation problem [MP14, Zha15, BBC
+
01], and sym-
metric oracle discrimination [BCMP16]. In addition, the coset identification problem we study gen-
eralizes the homomorphism evaluation problem for abelian groups studied by Zhandry in [Zha15],
which greatly inspired us. Section 7 gives details of this connection.
Daniel Copeland: daniel.copeland@gmail.com
Jamie Pommersheim: jamie@reed.edu
Accepted in Quantum 2021-03-03, click title to verify. Published under CC-BY 4.0. 1
arXiv:1812.09428v3 [cs.CC] 3 Mar 2021
In this paper, we analyze the query complexity of the general coset identification problem. We
prove that nonadaptive algorithms are optimal for any coset identification problem. We provide
tools to reduce the analysis of query complexity to purely character theoretic questions (which
are themselves often combinatorial). In particular we derive a formula for the exact quantum
query complexity for coset identification in terms of characters. In the case of symmetric oracle
discrimination (which itself includes polynomial interpolation as a special case) we find the lower
and upper bound for bounded error query complexity.
Another motivation for our work is the study of nonabelian oracles. Much is known about
quantum speedups when the oracle is a standard Boolean oracle. Less is known about whether
oracle problems with nonabelian symmetries can offer notable speedups. To that end we study the
follow scenario: suppose a group G acts by permutations on a finite set Ω (we call Ω a G-set). A
learner is given access to a machine which takes an element ω ∈ Ω and returns a·ω for some hidden
group element a ∈ G. With as few queries as possible the learner should guess the hidden element
a ∈ G. The classical query complexity for this problem is a long-known invariant of G-sets called
the base size. For instance, if G is the full permutation group of Ω = {1, . . . , n} then n −1 queries
are required classically to determine the hidden permutation. This problem is a special case of
symmetric oracle discrimination and we can express the bounded error quantum query complexity
of this purely in terms of the character of the G-set Ω. For instance, we find that when G is the full
permutation group of X = {1, . . . , n} then n−2
√
n+Θ(n
1/6
) queries are necessary (and sufficient)
to determine the hidden element.
This result bears some similarity to other work on learning problems related to the symmet-
ric group. Aaronson and Ambainis [AA14], who prove that at most a polynomial speedup can
be achieved in computing functions on n inputs which are invariant under the action of the full
symmetric group S
n
(using a standard evaluation oracle). Ben-David [BD16] proves that at most
a polynomial speedup is possible for Boolean functions defined on the full symmetric group. More
recently, Dafni, Filmus, Lifshitz, Lindzay and Vinyals [DFL
+
21] have studied the query complex-
ity of Boolean functions defined on the symmetric group, again proving a polynomial relationship
between the quantum and classical query complexities (as well as numerous other complexity mea-
sures). These results may be compared to the well-known fact that only polynomial speedups are
possible in computing total Boolean functions [BBC
+
01], the idea being that learning problems
on the full symmetric group correspond to total functions, while learning problems on a subgroup
correspond to partial functions. All of the results mentioned above are not directly comparable to
ours, since they use a standard evaluation oracle, while we examine a more symmetric “in-place”
oracle model.
The task of oracle identification can be further refined: fix a group G, a G-set Ω, and a func-
tion f : G → X which is constant on left cosets of some subgroup H, and distinct on distinct
cosets. The (left) coset identification problem is to determine f(a) given access to a permutational
black-box hiding a through the action on Ω. For instance, when G = S
n
(the symmetric group),
Ω = {1, . . . , n} its defining representation and f the sign homomorphism, it requires n −1 classical
queries to determine f(a). As a counterpoint to the harsh lower bound above we provide a family
of examples for this task parametrized by n in which the quantum query complexity is 1 while the
classical complexity is O(n). The groups we use are Heisenberg groups acting as small subgroups of
the full permutation group. This example is a nonabelian analogue of the fact that good quantum
speedups can be found in computing partial Boolean functions [BV97].
The paper is organized as follows. In section 2 we formalize coset identification in the context
of quantum learning algorithms and review the notions of adaptive and nonadaptive learning. In
section 3 we prove that parallel queries suffice to produce an optimal algorithm for this task. Sec-
tion 4 applies this theorem to symmetric oracle discrimination and addresses numerous example
problems. In section 5 we return to the general coset identification task and we prove the main
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theorem of this paper, Theorem 5.1, which is a formula for the success probability of an optimal t-
query algorithm in terms of characters. We use this in section 6 to compute the exact and bounded
error query complexity of some special examples (including the Heisenberg group example). We
conclude in section 7 by explaining how our work reproduces several previously known results
involving abelian oracles.
Our paper uses the language of representation theory of finite groups. A suitable reference is
the first third of Serre’s textbook [Ser96]. We review some important notations later in Section
5 (in particular, the idea of induced representation is critical for the statement of our results.)
Here we mention that a representation of a finite group G always refers to a finite dimensional and
unitary representation of G over the complex numbers. In other words, a representation is a group
homomorphism π : G → U (V ) (the unitary group of a f.d. vector space V ). We often think of V
as a left module for the group alegbra CG, and use the notation gv for π(g)v when the map π is
clear from the context.
2 Quantum learning from oracles
A quantum or classical oracle problem is described by a set of hidden information Y , a function
f : Y → X (the function to learn or compute), and a representation of Y as operations on inputs
of some kind (which determines the oracles). Classically such a representation consists of a set of
inputs Ω and an assignment taking each y ∈ Y to a permutation of Ω, i.e. a map π : Y → Sym(Ω).
A classical oracle problem is specified by a tuple (Y, Ω, π, f). A classical computer has access to
π(y) for some unknown y ∈ Y by spending one query to input ω ∈ Ω to learn π(y)·ω. The goal is to
determine f(y) with a high degree of certainty with as few queries as possible. More concretely, we
measure the efficacy of an algorithm by its average case success probability, namely the probability
of correctly outputting f(y) supposing the hidden information y is sampled uniformly from Y . For
the highly symmetric problems considered in this paper, this is the same as the worst-case success
probability, as explained below.
The quantum representation of oracles is described by a Hilbert space V and an assignment
taking each y ∈ Y to a unitary operator of V , in other words a map π : Y → U(V ). Thus a
quantum oracle problem is specified by a tuple (Y, V, π, f). The quantum computer spends one
query to input a state |ψi ∈ V to π(y) to acquire the state π(y)|ψi; the goal is to produce a state
and measurement scheme which outputs the value f(y).
Any classical oracle problem (Y, Ω, π, f) determines a quantum oracle problem via linearization:
oracles will act on the Hilbert space CΩ (spanned by the orthonormal basis {|ωi | ω ∈ Ω}) by
permutation matrices.
We note that there are other oracle models used to encode permutations. One possibility is
to require an oracle to act on a bipartite system, with one subsystem specified to be the “input
register” and the other a “response register”.
1
While we do not specifically consider this model
here, we note that many oracle problems, such as polynomial interpolation and group summation,
that are normally formulated in this two-register setup do have an easy reformulation in our setup.
Thus, our results and analyses apply to these problems in their original two-register formulation.
See Section 7. However, in some cases, the two-register setup results in a set of oracles that do not
form a group, for instance in the work of Ambainis on permutation inversion [Amb02]. In general,
it is an interesting question (and to our knowledge, open) whether these oracle models are the
1
More precisely, one usually defines an abelian group structure on Ω (usually cyclic) by defining an operation ⊕
on Ω. Then the oracle hiding the permutation π(a) is defined to act by |ω, bi 7→ |ω, (π(a) · ω) ⊕ bi. Here ω, b ∈ Ω,
so both the input and response registers are copies of CΩ.
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same, or if they lead to different query complexities.
2
A symmetric oracle problem is an oracle problem in which the hidden information is a group G
(so we are replacing Y with G) and the map π is a homomorphism G → Sym(Ω) in the classical
case or G → U(V ) in the quantum case. If π : G → U (V ) is a homomorphism, then it is common
practice to regard V as a (left) CG-module where CG is the group algebra of G (spanned by an
orthonormal basis sometimes written without kets as {g | g ∈ G}. In module notation we some-
times write g · v := π(g)(v) (for g ∈ G, v ∈ V ) if the representation π is understood from context.
The quantum oracle arising from a symmetric classical problem is also symmetric.
Of special interest to us is the case when the function f to be learned is compatible with the
group structure G. An instance of the coset identification problem is a symmetric oracle problem
(G, V, π, f) where the function f : G → X is constant on left cosets of a subgroup H ≤ G and dis-
tinct on distinct cosets. We also assume f is onto. The typical example is when X = {gH | g ∈ G}
is the set of left cosets of H and f(g) = gH. An equivalent formulation is to say that X is a
transitive G-set and the map f : G → X is a map of (left) G-sets (i.e., f(gh) = gf(h) for all
g, h ∈ G). Then the subgroup H can be recovered as the preimage of f(e).
For our analysis of the coset identification problem, we focus on average case success probabil-
ity. The symmetry of the problem implies that worst case and average case success probabilites
are equal, as the following argument shows: provided an unknown oracle π(a) we can select g ∈ G
uniformly at random and preprocess our input by applying π(g). Then an optimal average-case
algorithm will return the coset containing ga with optimal average-case success probability. The
coset which contains a can then be retrieved by applying g
−1
. Hence it suffices to consider the
average case success probability of any algorithm for this task (with the unknown oracle π(a) sam-
pled uniformly from G).
We examine bounded error and exact measures of query complexity. The exact (or zero error)
query complexity of a learning problem is the minimum number of queries needed by an algorithm
to compute f(y) with zero probability of error. The bounded error query complexity is the minimum
number of queries needed by an algorithm to compute f(y) with probability ≥ 2/3. The bounded
error query complexity is often studied for a family of problems growing with a parameter n and
so changing the constant 2/3 above to any number strictly greater than 1/2 will only change the
query complexity by a constant factor mostly ignored in asymptotic analysis.
Broadly speaking, there are two qualitatively different approaches to solving an oracle prob-
lem. The first approach is to ask questions one at a time, carefully changing your questions as you
receive more information. This is called using adaptive queries. The other approach is to prepare
all your questions and ask them at once in one go (imagining the learner has access to multiple
copies of the teacher). This is known as using non-adaptive, or parallel queries.
Classically the adaptive model is at least as strong as the nonadaptive model, since you can
convert any nonadaptive algorithm into an adaptive one (by picking your questions in advance but
asking them one at a time). This is well-known to be true also in the quantum setting. In the next
section we will prove the converse for coset identification:
Theorem 2.1. Suppose (G, V, π, f) describes an instance of coset identification. Then there exists
a t-query quantum algorithm to determine f(a) with probability P if and only if there exists a
t-query nonadaptive query algorithm which does the same.
This theorem is certainly not true for arbitrary learning problems: Grover’s algorithm provides
2
As another modification, one may propose that having access to an oracle O means an algorithm may choose
to access O or O
−1
in any given query. This is a separate model which we do not consider here.
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an example in which any optimal algorithm must use adaptive queries [Zal99]. To prove the
theorem we must precisely state what adaptive and nonadaptive algorithms are.
2.1 Adaptive vs. nonadaptive: definitions
Recall that a quantum learning problem is described by a tuple (Y, V, π, f : Y → X) where Y
indexes the set of hidden information, V is a finite dimensional Hilbert space, π : Y → U(V ) a
representation of the unknown information by unitary operators, and f is the function to learn.
The standard model for an adaptive algorithm is as follows (see e.g. [BBC
+
01, Section 3.2]):
A t-query adaptive quantum algorithm for the quantum oracle problem (Y, V, π, f : Y → X)
consists of a tuple A = (N, ψ, {U
1
, . . . , U
t
}, {E
i
}) where
• N is the dimension of the auxiliary workspace used in the computation
• |ψi is a unit vector in V ⊗ C
N
• {U
1
, . . . , U
t
} is a set of unitary operators acting on V ⊗ C
N
• {E
x
}
x∈X
is a POVM with measurement outcomes indexed by X.
The algorithm uses t queries to the oracle π(a) (with a sampled uniformly from Y ) to produce
the output state
|ψ
A
a
i = U
t
(π(a) ⊗ I)U
t−1
(π(a) ⊗ I) . . . (π(a) ⊗ I)U
1
(π(a) ⊗ I)|ψi
upon which the algorithm executes the measurement described by {E
x
}
x∈X
. Here and elsewhere
I denotes the identity operator (in this case acting on the space C
N
).
In quantum circuit notation the preparation of the state |ψ
A
a
i reads:
|ψi
π(a)
U
1
π(a)
. . .
π(a)
U
t−1
π(a)
U
t
= |ψ
A
a
i
By contrast, an algorithm is nonadaptive if at any point during the algorithm, the input for
some query does not depend on the results to any of the previous queries. Essentially this means
that all the inputs are completely determined before the algorithm begins. Classically, t nonadap-
tive queries are identical to t simultaneous queries to t copies of an oracle. This motivates the
following definition (cf [Mon10, Section 2]):
A t-query nonadaptive quantum algorithm for the oracle problem (Y, V, π, f) is a tuple A =
(N, ψ, {E
x
})
x∈X
where
• N is the dimension of the auxiliary register.
• |ψi is the input state, a unit vector of V
⊗t
⊗ C
N
.
• {E
x
} is a POVM indexed by X.
The algorithm operates on the input state to produce
|ψ
A
a
i = (π(a)
⊗t
⊗ I)|ψi
which is then measured using the POVM {E
x
}. The next fact is very useful and follows immediately
from definitions.
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Lemma 2.2. A t-query nonadaptive algorithm for the problem (Y, V, π, f) is the same as a single-
query nonadaptive algorithm for the oracle problem (Y, V
⊗t
, π
⊗t
, f).
The quantum circuit notation for the nonadaptive preparation of the state |ψ
A
a
i is drawn as
follows.
|ψi
π(a)
π(a)
.
.
.
π(a)
= |ψ
A
a
i
In either model, the algorithm A uses t copies of the unitary π(a) to produce a state |ψ
A
a
i.
Using the POVM {E
x
} results in a measurement value x ∈ X with probability
P (x | a) = hψ
A
a
|E
x
|ψ
A
a
i.
Since we assume the oracle is sampled uniformly from Y , the probability that A executes
successfully is
P
succ
(A) =
1
|Y |
X
a∈Y
P (f(a) | a) =
1
|Y |
X
a∈Y
hψ
A
a
|E
f(a)
|ψ
A
a
i.
2.2 Symmetric oracle problems
Suppose we have a symmetric oracle problem (G, V, π, f ). As mentioned in the introduction, since
we are focusing on query complexity and not on issues of implementation, analysis of this problem
depends only on the character χ
V
of π : G → U(V ), as we prove in the lemma below. In fact, a
little more is true. Let Irr(G) denote the set of irreducible characters of G. Given a representation
π : G → U(V ) define the set
I(V ) := {χ ∈ Irr(G) appearing in the representation V }
= {χ ∈ Irr(G) | (χ, χ
V
) > 0}.
Here we are using (·, ·) to denote the usual inner product of characters. If χ ∈ Irr(G) and (χ, χ
V
) >
0 we say that χ appears in the representation V .
Lemma 2.3. The optimal success probability of a t-query algorithm to solve a symmetric oracle
problem (G, V, π, f) depends only on I(V ) and f .
Proof. First, note that if U : V → W is a Hilbert space isomorphism then we can define a new
oracle problem (G, W, UπU
−1
, f) where the oracles now act on W . Any t-query algorithm to solve
the original problem can be “conjugated” by U (e.g. the input state |ψi becomes U|ψi and the
non-oracle unitaries and POVM are conjugated by U) to produce a t-query algorithm for the new
problem which succeeds with the same probability. Conversely any algorithm to solve the new
problem can be conjugated by U
−1
to solve the old problem with the same probability. Therefore
oracle problems with isomorphic unitary representations of G will have the same t-query optimal
success probability. In other words, only the character χ
V
is relevant.
Second, we claim that the multiplicities of irreducible characters in V are not important; only
whether they appear in V or not. Indeed, adding a d-dimensional workspace to a computer’s
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original system V produces a new representation V ⊗ C
d
of G with character dχ
V
. Since we
allow our algorithm to introduce any such workspace, we are in effect allowing it to increase the
multiplicity of each character by a factor of d. Note that this process will never produce irreps
which did not appear in V to begin with. Hence the optimal success probability depends only on
which irreps appear in V , i.e. the set I(V ).
It makes sense that if an algorithm is granted access to more representations to work with, its
success probability cannot decrease. To be more precise, fix t, and let P
opt
(G, V, π, f) denote the
optimal success probability of a t-query algorithm for the symmetric oracle problem (G, V, π, f ).
Lemma 2.4. Suppose π
V
, π
W
are representations of G on the spaces V and W , with I(W ) ⊂ I(V ).
Then
P
opt
(G, W, π
W
, f) ≤ P
opt
(G, V, π
V
, f).
Proof. The basic idea is any t-query algorithm to solve (G, W, π
W
, f) can be extended to produce a
t-query algorithm for (G, V, π
V
, f). Suppose an algorithm A for W uses an N dimensional ancilla
space, i.e. operates on W ⊗ C
N
. Since I(W ) ⊂ I(V ), there exists some M so that V ⊗ C
M
contains a subrepresentation isomorphic to W ⊗ C
N
. Hence we can write V ⊗ C
M
= W
0
⊕ Y
where W
0
∼
=
W ⊗ C
N
as CG-modules. Now we claim the initial state, intermediate unitaries, and
POVM for the algorithm A can be extended to an algorithm A
0
acting on V ⊗ C
M
. The initial
state for A
0
is the vector in W
0
⊂ V ⊗ C
N
corresponding to the initial state for A in W ⊗ C
N
.
The intermediate unitaries for A
0
act on W
0
according to the unitaries for the algorithm A and
are extended arbitrarily to Y . The measurement operators for A
0
all agree with the measurement
operators for A on W
0
and all but one of the operators act as 0 on the subspace Y . To satisfy the
completeness relation on V ⊗C
M
, exactly one of the POVMs should act as the identity on Y (this
modification is unimportant since A
0
“takes place” entirely within W
0
). The success probability of
A
0
is equal to that of A.
3 Parallel queries suffice
Here we prove Theorem 2.1, namely that the optimal success probability for coset identification
can be attained by a parallel (nonadaptive) algorithm. We prove this by showing that any t-query
adaptive algorithm can be converted to a t-query nonadaptive algorithm without affecting the suc-
cess probability. Another way to say this is that every t-query adaptive algorithm can be simulated
by a t-query nonadaptive one. This technique is greatly inspired by the work of Zhandry [Zha15]
who proves this result when G is abelian, and also bears resemblance to the lower bound technique
of Childs, van Dam, Hung and Shparlinski [CvDHS16], where the special case of polynomial inter-
polation is addressed.
Let π : G → U(V ) be a unitary representation of G. Let CG denote the group algebra of
G. Each h ∈ G acts on CG by left multiplication, an operator we denote L
h
. We will use the
controlled multiplication operator ([DBCW
+
02]) defined on V ⊗ CG by
CM|v, gi = |π(g
−1
)v, gi.
This defines a unitary operator and is a generalization of the standard CNOT gate (take G = Z
2
and V = CZ
2
). As such we draw it using circuit diagrams as in section 3.
V
CG
Figure 1: Notation for the controlled multiplication gate CM .
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There are two G-actions on V ⊗ CG we use, one given by π(h) ⊗ L
h
and the other I ⊗ L
h
. Our
first observation is that CM intertwines these actions.
Lemma 3.1. The controlled multiplication operator satisfies
CM(π(h) ⊗ L
h
) = (I ⊗ L
h
)CM.
The proof follows by applying both sides to a vector |v, gi and using the definition of CM. The
representation obtained by letting each h ∈ G act by the identity on V is a direct sum of dim V
many copies of the trivial reprseentation, so we denote it
⊕ dim V
. The lemma allows us to in-
terpret CM as a CG-module isomorphism V ⊗CG →
⊕ dim V
⊗CG. In pictures the lemma reads:
π(h)
L
h
L
h
=
Figure 2: Lemma 3.1 in pictures.
The next property is crucial for our parallelization argument. Recall that if W is a CG-module
then I(W ) denotes the set of irreducible characters of G which appear in W .
Lemma 3.2. Suppose W is a subrepresentation of CG. Then there is a subrepresentation Y of CG
such that the image of V ⊗W under CM is contained in V ⊗Y and Y satisfies I(Y ) = I(V ⊗W ).
Proof. By Lemma 3.1 CM is a CG-module isomorphism V ⊗ CG →
⊕ dim V
⊗ CG where V and
⊕ dim V
have the same underlying vector space. Let Z denote the image of V ⊗ W under CM.
Then CM restricts to a CG-module isomorphism V ⊗ W → Z. Next let Y be the submodule of
CG which contains each irreducible of I(Z) with maximal multiplicity (so if χ appears in Y then χ
appears with multiplicity χ(e)). Now Z
∼
=
V ⊗W as CG-modules so in particular I(Z) = I(V ⊗W).
Hence also I(Y ) = I(V ⊗ W ).
It remains to prove Z ⊆ V ⊗ Y . Indeed, in the CG-module
⊕ dim V
⊗ CG the subspace
⊕ dim V
⊗Y is the maximal subrepresentation containing only irreducibles in I(V ⊗W ). As noted
Z contains only irreducibles in I(V ⊗ W ) so therefore Z ⊆
⊕ dim V
⊗Y , which is the same vector
space as V ⊗ Y .
Now suppose (G, V, π, f) is an instance of coset identification and A =
(N, |ψi, {U
1
, . . . , U
t
}, {E
x
}) is a t-query adaptive algorithm to evaluate the homomorphism
f. First, by replacing π with π ⊗I if necessary, we may assume that the algorithm does not use a
workspace, that is N = 1. We will describe a new adaptive algorithm A
0
which is a modification
of A as follows. We introduce a new workspace which is a copy of CG. The new intermediate
unitaries are (U
1
⊗ I)CM, (U
2
⊗ I)CM, . . . , (U
t
⊗ I)CM. The input state is |ψi ⊗ |ηi where η is
the equal superposition state in CG. When the oracle is hiding the unitary π(a) this produces the
following state:
|ψi
|ηi
π(a)
U
1
π(a)
. . .
U
t−1
π(a)
U
t
Figure 3: Pre-measurement state for A
0
.
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Next measurement is performed: first the second register is measured in the standard basis of
CG. Then the original POVM is applied to the first register. The result of these two measurements
will be a pair (g, x); the final output of the algorithm is gx.
3
Lemma 3.3. The algorithm A
0
succeeds with the same success probability as A.
Lemma 3.4. The algorithm A
0
can be simulated by a t-query parallel query algorithm.
Proof of Theorem 2.1 from Lemmas 3.3 and 3.4. By the two lemmas, given any t-query adaptive
algorithm A which solves coset identification with probability P , there exists a t-query parallel
query algorithm which succeeds with the same probability.
Proof of Lemma 3.3. Consider the pre-measurement state for A
0
given that the hidden group ele-
ment is a ∈ G. It can be written
|ψ
A
0
a
i =
1
p
|G|
X
g∈G
|ψ
A
g
−1
a
i ⊗ |gi.
If the first measurement reads g then the state collapses to |ψ
A
g
−1
a
i⊗|gi. If the second measurement
is now performed, the result will read f(g
−1
a) with the same probability that the algorithm A
would read this result given that the oracle was hiding g
−1
a. The algorithm then classically
converts the result to gf (g
−1
a) which is equal to f(a) since f is a left G-set map. So the following
conditional probabilities are equal:
P (A
0
outputs f(a) | a is hidden , first measurement result is g)
= P (A outputs f(g
−1
(a)) | g
−1
a is hidden ).
Denote these probabilities by P
A
0
(f(a) | a, g) and P
A
(f(g
−1
a) | g
−1
a) respectively. Since the prob-
ability that the first measurement of A
0
reads g is 1/|G| for all G and g is sampled independently
of a, we compute the average case success probability by
P
succ
(A
0
) =
1
|G|
2
X
g∈G
X
a∈G
P
A
0
(f(a) | a, g)
=
1
|G|
2
X
g∈G
X
a∈G
P
A
(f(g
−1
a) | g
−1
a)
=
1
|G|
X
g∈G
P
succ
(A) = P
succ
(A).
Proof of Lemma 3.4. We rewrite the pre-measurement state of A
0
expressed by Figure 3 using
Lemma 3.1. Denote the state that results when the hidden element is a ∈ G by |ψ
A
0
a
i. We apply
Lemma 3.1 diagrammatically from left to right:
|ψi
|ηi
|ψ
A
0
a
i =
π(a)
U
1
π(a)
. . .
U
t−1
π(a)
U
t
3
Formally the algorithm A
0
is given by
A
0
= (|G|, |ψ, ηi, {U
1
⊗ I ◦ CM, . . . , U
t
⊗ I ◦ CM}, {E
0
x
=
X
g∈G
E
g
−1
x
⊗ |gihg|}).
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|ψi
|ηi
=
L
a
−1
U
1
π(a)
L
a
. . .
U
t−1
π(a)
U
t
|ψi
|ηi
=
L
a
−1
U
1
L
a
. . .
U
t−1
π(a)
U
t
U
2
|ψi
|ηi
· · · =
L
a
−1
U
1
. . .
U
t−1
π(a)
U
t
U
2
L
a
|ψi
|ηi
=
U
1
. . .
U
t−1
π(a)
U
t
U
2
L
a
In the last step, in addition to applying Lemma 3.1 at the right of the diagram, we used the
fact that L
a
−1
|ηi = |ηi. In formulas we have
|ψ
A
0
a
i = (I ⊗ L
a
) ◦
(U
t
⊗ I) ◦ CM ◦··· ◦ (U
1
⊗ I) ◦ CM
|ψ, ηi.
Therefore we have converted this algorithm to a single-query algorithm using the oracle I ⊗ L
a
with initial state U |ψ, ηi where U = (U
t
⊗ I) ◦ CM ◦··· ◦ (U
1
⊗ I) ◦ CM.
Claim. The image of V ⊗ C|ηi under U is contained in V ⊗ Y where Y ⊆ CG is a submodule
satisfying I(Y ) = I(V
⊗t
).
This is readily proved by induction and Lemma 3.2. For instance, by Lemma 3.2 the image of
V ⊗C|ηi under CM is contained in V ⊗Y
1
where Y
1
is a submodule with I(Y
1
) = I(V ). The next
part of U is U
1
⊗ I which sends V ⊗ Y
1
to itself. Now another CM is applied and by Lemma 3.2
this sends V ⊗ Y
1
to V ⊗ Y
2
where I(Y
2
) = I(V ⊗ Y
1
) = I(V
⊗2
).
Therefore the inital state U|ψ, ηi belongs to the subspace V ⊗Y , which means that the algorithm
A
0
may be simulated by a single query algorithm to the oracle I ⊗L
a
acting on the subspace V ⊗Y .
Note that the irreducibles appearing in this subspace are I(
⊕ dim V
⊗ Y ) = I(Y ) = I(V
⊗t
).
Hence Lemma 2.3 implies there exists a single-query algorithm using the representation V
⊗t
which
achieves the same success probability as A
0
. As noted in Lemma 2.2 this is the same as a t-query
parallel algorithm using the representation V . This concludes the proof of Lemma 3.4.
Corollary 3.5. The optimal t-query success probability for an algorithm solving an instance of
coset identification (G, V, π, f) is equal to the optimal single-query success probability achievable
solving the instance (G, V
⊗t
, π
⊗t
, f).
4 Application to symmetric oracle identification
Symmetric oracle discrimination is the following task: given oracle access to a symmetric oracle
hiding a group element a ∈ G, determine a exactly. This is the special case of coset identifi-
cation in which H = {e}. Thus an instance of this problem is determined by a finite group G
and a (finite-dim) unitary representation π : G → U (V ). The following theorem computes the
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success probability of a single-query algorithm and is proved by Bucicovschi, Copeland, Meyer and
Pommersheim:
Theorem 4.1. ([BCMP16], Theorem 1) Suppose G is a finite group and π : G → U(V ) a unitary
representation of G. Then an optimal single-query algorithm to solve symmetric oracle discrimi-
nation succeeds with probability
P
opt
=
d
V
|G|
where
d
V
=
X
χ∈I(V )
χ(e)
2
.
The result of the previous section tells us that parallel algorithms are optimal for symmetric
oracle discrimination.
Theorem 4.2. Suppose G is a finite group and π : G → U(V ) a unitary representation of G. Then
an optimal t-query algorithm to solve symmetric oracle discrimination succeeds with probability
P
opt
=
d
V
⊗t
|G|
where
d
V
⊗t
=
X
χ∈I(V
⊗t
)
χ(e)
2
.
Proof. Theorem 2.1 tells us that a t-query parallel algorithm achieves the optimal success proba-
bility. As noted this is equivalent to a single-query algorithm using the representation π
⊗t
: G →
U(V
⊗t
). Now apply Theorem 4.1.
To express the exact and bounded error query complexity of symmetric oracle discrimination
we’re compelled to make the following definitions.
Let V denote a CG-module. The quantum base size, denoted γ(V ), is the minimum t for which
every irrep of G appears in V
⊗t
. If no such t exists then γ(V ) = ∞. The bounded error quantum
base size, denoted γ
bdd
(V ) is the minimum t for which
1
|G|
X
χ∈I(V
⊗t
)
χ(e)
2
≥ 2/3.
If (G, V, π) is a case of symmetric oracle discrimination then by Theorem 4.2 the number of
queries needed to produce a probability 1 algorithm is γ(V ). That is, the exact quantum query
complexity of the problem is equal to the quantum base size of V . Similarly the bounded error
query complexity is γ
bdd
(V ).
It may happen that one of these quantities is infinite. However when V is a faithful represen-
tation then a classical result attributed to Brauer and Burnside ([Isa76], Theorem 4.3) guarantees
that every irrep of G appears in one of the tensor powers V
⊗0
, V, V
⊗2
, . . . , V
⊗m−1
where m is the
number of distinct values of the character of V . If V contains a copy of the trivial representation,
then we can say that every irrep of G is contained in some tensor power V
⊗t
for some t. Hence in
this case (with V faithful and containing a copy of the trivial irrep) both γ(V ) and γ
bdd
(V ) are
finite.
In particular, this occurs whenever we “quantize” a classical symmetric oracle discrimina-
tion problem. This is the learning problem specified by a finite set Ω and a homomorphism
G → Sym(Ω). A query to an oracle hiding a ∈ G consists of inputting ω ∈ Ω and receiving
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a · ω. The learner must determine the hidden group element (or permutation) a. The quantized
learning problem uses the homomorphism G → U(CΩ) sending elements of G to permutation
matrices. (Such a representation is called a permutation representation.) Then the quantized
learning problem is faithful if the original problem is faithful and the CG-module contains a copy
of the trivial representation, namely span{
P
ω∈Ω
|ωi}.
This is precisely the situation we would like to study because we can compare the classical and
quantum query complexity. Classically the exact and bounded error query complexities are equal,
since if a classical algorithm does not use enough queries to identify the hidden permutation with
certainty then it must make a guess between at least 2 equally likely permutations which behave
the same on all the queries that were used, resulting in a success rate of at most 1/2.
• Suppose Ω = {1, . . . , n} hosts the defining permutation representation of G = S
n
. Then
n − 1 queries are required to determine a hidden permutation σ.
• If we take the same action but restrict the group to A
n
≤ S
n
then we need n − 2 queries to
determine a hidden element σ ∈ A
n
.
• Consider the action of the dihedral group D
n
on the set of vertices of an n-gon. Then 2
queries are required to determine a hidden group element.
In general the classical query complexity is a well-known invariant of a permutation group G
denoted b(G) called the minimal base size or just base size of G [LS:02]. It may be defined to be the
length of the smallest tuple (ω
1
, . . . , ω
t
) ∈ Ω
t
with the property that (g·ω
1
, . . . , g·ω
t
) = (ω
1
, . . . , ω
t
)
if and only if g = 1. From the definition it is clear that the base size agrees with the non-adaptive
classical query complexity of the problem. In fact, it is also equal to the adaptive query complexity,
since if a sequence of adaptive guesses (ω
1
, . . . , ω
t
) suffices to identify a particular hidden g ∈ G,
then the same sequence of guesses works for every element of the group. This means any optimal
algorithm may be implemented non-adaptively. Thus the classical query complexity of symmetric
oracle discrimination of G ≤ Sym(Ω) is the base size of G and the quantum exact (bounded error)
query complexity is the (bounded error) quantum base size. We are naturally led to a broad
group theoretic problem:
Question. What are the relationships between b(G), γ(CΩ) and γ
bdd
(CΩ)?
We are not aware of any direct comparison of these quantities in the group theory literature.
Here we only compute the various quantities for some special cases. We saw earlier that b(S
n
) =
n − 1. We will prove
Theorem 4.3. Let γ, γ
bdd
denote the quantum base sizes for S
n
acting on {1, . . . , n}. Then
1. γ = n − 1 queries are necessary and sufficient for exact learning.
2. γ
bdd
= n − 2
√
n + Θ(n
1/6
) queries are necessary and sufficient to succeed with probability
2/3.
3. In fact, for any ∈ (0, 1), n −2
√
n + Θ(n
1/6
) queries are necessary and sufficient to succeed
with probability 1 − .
Proof. Recall that the irreducible characters of S
n
are parametrized by partitions of n which can
be written either as a sequnce [λ
1
, . . . , λ
n
] or as a Young diagram with n total boxes and λ
i
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boxes in the ith row. Let V = C{1, . . . , n} denote the CG-module corresponding to the defining
permutation representation of S
n
. Then V decomposes as a sum of two irreducibles:
V = V
[n]
⊕ V
[n−1,1]
.
We note that V
[n]
is the trivial representation. A well-known rule says that if V
λ
is a simple
representation corresponding to the Young diagram λ then the irreps appearing in V ⊗ V
λ
I(V ⊗ V
λ
) = {V
µ
| µ ∈ λ
±
}.
where λ
±
is the set of Young diagrams obtained from λ by adding then removing a box from
lambda. In particular, this shows by induction that
I(V
⊗t
) = {V
µ
| µ has at least n − t columns}.
We see that n − 1 queries are required until every irreducible is contained in V
⊗t
(in particular,
the sign representation corresponding to the partition [1
n
] = [1, 1, . . . , 1] is not included in V
⊗t
unless t ≥ n −1). This proves part (1) of the theorem.
To prove part (2) we must examine more closely the set I
t
= I(V
⊗t
) consisting of all partitions
with at least n − i columns (i.e. λ
1
≥ n −i). We are interested in the sum
d
t
:= d
V
⊗t
=
X
χ∈I(V
⊗t
)
χ(e)
2
.
It is well known that if χ is an irrep corresponding to the Young diagram λ then χ(e) is equal
to the number of standard tableaux of shape λ ([Sag01], Theorem 2.5.2). Hence χ(e)
2
is equal
to the number of pairs of standard tableaux of shape λ. Now by the Robinson-Schensted corre-
spondence, the sum above is equal to the number of sequences of the numbers {1, . . . , n} whose
longest increasing subsequence is at least n − t (see e.g. [Sag01], Theorem 3.3.2). Next, a deep
result of Baik, Deift and Johannson [BDJ99] identifies the distribution of the l
n
, the length of
the longest increasing subsequence of a random permutation of n elements, as the Tracy-Widom
distribution (which also governs the largest eigenvalue of a random Hermitian matrix) of mean
2
√
n and standard deviation n
1/6
. In particular, Theorem 1.1 of [BDJ99] asserts that if F (x) is
the cumulative distribution function for the Tracy-Widom distribution, then
lim
n→∞
P rob
l
n
− 2
√
n
n
1/6
≤ x
= F (x)
Let c be any real number. If we use t = n −2
√
n + cn
1/6
queries, then our success probability will
be
P rob(l
n
≥ n −t) = 1 − P rob(l
n
< 2
√
n − cn
1/6
) = 1 − P rob
l
n
− 2
√
n
n
1/6
< −c
→ 1 −F (−c)
Thus for any ∈ (0, 1), if we wish to succeed with probability 1 − , it will be necessary and
sufficient to use t = n − 2
√
n + cn
1/6
queries, where c = −F
−1
() (for n sufficiently large).
Here is the analogous result for identifying an element of the alternating group.
Theorem 4.4. Consider the standard action of A
n
acting on {1, . . . , n}. Then the quantum base
sizes are given as follows.
1. γ = n − d
√
ne are necessary for exact learning.
2. γ
bdd
= n−2
√
n+Θ(n
1/6
) are necessary and sufficient to succeed with probability 2/3. In fact,
for any ∈ (0, 1), n −2
√
n + Θ(n
1/6
) are necessary and sufficient to succeed with probability
1 − .
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Proof. Recall the following facts about the representation theory of A
n
. The conjugate of a par-
tition λ is the partition λ
∗
obtained by swapping the rows and columns of λ; in other words
λ
∗
= (λ
∗
1
, λ
∗
2
, . . . ) where λ
∗
i
= the number of boxes in the ith column of λ. For each partition λ of
n that is not self-conjugate, i.e. λ 6= λ
∗
, the restriction of V
λ
to A
n
is an irreducible representation
W
λ
of A
n
. Also, W
λ
= W
λ
∗
. For self conjugate λ, the representation V
λ
breaks up into two
distinct irreducible representations W
+
λ
and W
−
λ
of equal dimension.
Recall from the previous proof that after t queries, we get copies of all the V
λ
such that
λ
1
≥ n − t. Observe that for any partition λ, we must have either λ
1
≥ d
√
ne or λ
∗
1
≥ d
√
ne.
(If both fail, the partition fits into a square of side length d
√
ne − 1, which contains fewer than n
boxes.) It follows that after t = n − d
√
ne queries, for any λ, we have picked up a copy of V
λ
or
V
λ
∗
. Hence we have every irreducible representation of A
n
. Therefore, n − d
√
ne queries suffice
for exact learning. Showing that that fewer queries cannot suffice is similar. Here we make the
observation that there exists a partition λ such that λ
1
< d
√
ne + 1 and λ
∗
1
< d
√
ne + 1, since n
boxes can be packed into a square of side length d
√
ne. It follows that t = n − d
√
ne − 1 queries
do not pick up the V
λ
or V
λ
∗
for such λ. Thus, we do not get every irrep of A
n
.
We now examine the bounded error case. For a positive integer t, let p
t
be the success prob-
ability of the optimal t-query algorithm for identifying a permutation of S
n
and let q
t
be the
corresponding probability for A
n
.
Let V denote the t-fold tensor power of the defining representation of S
n
. We can decompose
V as a direct sum of irreps of S
n
and if we know which V
λ
appear we can determine which irreps
of A
n
appear in V . In particular, each time we have a non-self-conjugate λ such that V
λ
appears
in V , we will have W
λ
appearing in V . Let’s consider the contribution of this appearance to the
success probability p
t
and q
t
, which is the square of the dimension divided by the order of the
group. Since the dimension of V
λ
equals the dimension of W
λ
, while the order of S
n
is twice the
order of A
n
, the contribution to q
t
is twice the contribution to p
t
.
Now if λ is self-conjugate then V
λ
decomposes into two irreps of S
n
of equal dimensions. The
sum of the squares of these two irreps is thus one-half the square of the dimension of V
λ
. Once we’ve
divided by the sizes of the groups, we see that the contribution to q
t
is equal to the contribution
to p
t
.
We have thus seen that for any λ the contribution to q
t
is either 2 or 1 times the contribution
to p
t
. It follows that
p
t
≤ q
t
≤ 2p
t
Thus for q
t
≥ 2/3 we must have p
t
≥ 1/3, which as we showed in Theorem 4.3 requires n −2
√
n +
Θ(n
1/6
) queries. On the other hand, if we are given n−2
√
n+Θ(n
1/6
) queries, we achieve p
t
≥ 2/3,
which forces q
t
≥ 2/3.
The two theorems above show that there is very little speedup possible when trying to identify
a permutation from the symmetric group or the alternating group. For the alternating group, one
can at least get by with
√
n fewer queries for exact quantum learning. Here there is an analogy
to Van Dam’s problem of exactly learning the value of an n-long bitstring using queries to its bits
[van98]. Exact learning requires n queries. However, if we are guaranteed in advance that the
parity of the string is even, then only bn/2c queries are required for exact learning. To see this
using the techniques of the current paper, we argue as follows. Let G be the subgroup Z
n
2
consisting
of all strings of even parity. If we are allowed t queries, then we can access those representations ρ
x
of Z
n
2
corresponding to strings x of Hamming weight less than or equal to t (see also the remarks
in Section 7.3). If ¯x is the bitwise complement of x, then ρ
x
and ρ
¯x
take the same values on G.
Now, for any string x, one of x and ¯x will have Hamming weight less than or equal to bn/2c. Hence
every representation of G can be accessed by bn/2c queries to the oracle, and we will succeed with
probability 1.
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5 Query complexity of coset identification
In this section we derive a formula for the optimal success probability of a t-query algorithm
to solve coset identification. In light of our previous result on parallelizability (Corollary 3.5),
this boils down to finding a formula in the single-query case. This will directly generalize the
single-query results of [BCMP16] used in Section 4.
To state the result we fix some notation. Suppose (G, V, π, f) is an instance of coset identi-
fication with H the preimage of f (e). Given an H-representation W let W
↑
denote the induced
representation of W (which is a representation of G; see Section 5.1.1 below for more details.)
Likewise if W is a CG-module then we denote by W
↓
the CH-module obtained by restriction to
H. Recall that if V is a CG-module then I(V ) denotes the set of all irreducible characters of G
appearing in V . We sometimes use the notation I
G
(V ), I
H
(V ) to emphasize which group we are
considering. Finally, given two representations A and B we let
A
B
:= the maximal subrepresentation of A such that I(A
B
) ⊆ I(B).
Thus A
B
denotes the sum of all the isotypical components of A which correspond to an irreducible
isotype appearing in B. We will be interested in the quantities
dim A
B
dim A
which can be understood as the fraction of A which is shared with B.
Theorem 5.1. An optimal single-query algorithm to solve the instance (G, V, π, f) of coset iden-
tification succeeds with probability
P
opt
= max
Y ∈Irr(H)
dim (Y
↑
)
V
dim Y
↑
.
In words: to find the optimal success probability, you look at an irrep Y of H which appears
in V
↓
. Then you examine the fraction of Y
↑
which is shared with V . Finally take the maximum
over all irreps Y appearing in V
↓
.
From this theorem we can quickly deduce Theorem 4.2, the single-query result for symmetric
oracle identification. This is the special case when H is the trivial group. Then H has only one
irrep, namely the trivial representation , and
↑
is isomorphic to CG. Hence the formula we get
from Theorem 5.1 is
P
opt
=
dim(CG)
V
|G|
=
1
|G|
X
χ∈I(V )
χ(e)
2
which is the formula of Theorem 4.2.
The next two sections are devoted to the proof of Theorem 5.1. First we prove the lower bound
(i.e. existence of a state and measurement achieving the desired success probability) and then we
prove the upper bound (optimality of that success probability).
5.1 The lower bound
First we collect some facts concerning induced representations and averaging operators needed for
the proof of Theorem 5.1. A fine treatment of the subject is contained in Serre’s book [Ser96].
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5.1.1 Induced Representations
Suppose H is a subgroup of a finite group G and let Y denote a representation of H. Note that
CG admits a right H-action. The representation of G induced from Y is
Y
↑
G
H
= CG ⊗
CH
Y.
When H and G are understood we simply write Y
↑
. Similarly if W is a representation of G then it
is also a representation of H, called the restriction of W to H. We denote it by W
↓
G
H
or simply W
↓
.
From the definition of induced representations, we can write
Y
↑
=
M
t
t ⊗ Y
where t ranges over a set of coset representatives for H. Conversely, if a representation W of G
contains an H-invariant subspace W
0
such that
W =
M
t
tW
0
where t again ranges over a set of coset representatives for H, then W is isomorphic to W
↑
0
as G
representations.
In our situation all representations are unitary. In particular if Y is a unitary representation
of H then Y
↑
is equipped with the inner product determined by requiring the subspaces t ⊗ Y to
be pairwise orthogonal, and translating the inner product of Y to each subspace t ⊗ Y . With this
inner product Y
↑
is a unitary representation of G. We will often denote the orthogonal projection
onto e ⊗Y by E. Then the orthogonal projection onto t ⊗Y is tEt
−1
, and we have
P
t
tEt
−1
= I.
5.1.2 Averaging operators
Given a CG-module V we can define the averaging operator, which turns an arbitrary linear map
A : V → V into a G-invariant one:
R
G
: End
C
(V ) → End
G
(V )
R
G
(A) :=
1
|G|
X
g∈G
gAg
−1
.
Note that R
G
(A) commutes with every g ∈ G so that indeed R
G
(A) is G-invariant, i.e. R
G
(A) ∈
End
G
(V ). If B is a G-invariant operator then R
G
(BA) = BR
G
(A). The map R
G
is trace-
preserving, so in particular if p is a projection then R
G
(p) is non-zero, since it has non-zero trace.
If V contains only a single isotype of irrep, i.e. V
∼
=
Y ⊗ C
m
for some irrep Y then R
G
is closely
related to the partial trace with respect to the subspace Y :
R
G
(A) =
1
dim Y
I ⊗ Tr
Y
(A). (1)
5.1.3 Proof of the lower bound
Before giving the proof of the lower bound in Theorem 5.1 we prove a preliminary proposition.
If R is a algebra over C, V an R-module and W ≤ V a linear subspace, we let R · W denote
the submodule of V generated by W (i.e. the smallest submodule containing the subspace W ).
Similarly for r ∈ R we let r · W denote the subspace {rw : w ∈ W }.
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Proposition 5.2. Suppose Y is an irreducible unitary representation of H (a subgroup of G). Also
suppose V is a G-subrepresentation of Y
↑
. Let E denote orthogonal projection onto e ⊗ Y ⊂ Y
↑
.
Then there exists a unit vector ψ ∈ V such that
hψ|E|ψi =
dim V
dim Y
↑
.
Remark. In Proposition 5.7 we will prove this is an upper bound for hψ|E|ψi over all unit
vectors ψ ∈ V .
Proof. Let Π
V
denote the G-invariant orthogonal projection onto V . Since Y is irreducible, E is
a minimal idempotent in End
H
(Y
↑
). Therefore, since Π
V
also belongs to End
H
(Y
↑
), we know
EΠ
V
E is a scalar times E. In turn this implies Π
V
EΠ
V
is a scalar multiple of an orthogonal
projection, since it is self-adjoint and
(Π
V
EΠ
V
)
2
= Π
V
EΠ
V
EΠ
V
= scalar · Π
V
EΠ
V
.
The image of Π
V
EΠ
V
is an H-invariant subspace of V which is either 0 or isomorphic to Y .
Let this subspace be Y
0
, so we have
Π
V
EΠ
V
= λΠ
Y
0
(2)
for some non-zero scalar λ ∈ C. We will also use the fact that
Π
Y
0
EΠ
Y
0
= λΠ
Y
0
, (3)
which results from Eq. (2) by multiplying the equation by Π
Y
0
on the left and right. Next, we
claim that Y
0
is not zero (so it is in fact isomorphic to Y as an H-module). Indeed, we have
CG · Y
0
=
X
t
t · Y
0
=
X
t
Im(Π
V
tEt
−1
Π
V
) ⊃ Im(Π
V
(
X
t
tEt
−1
)Π
V
) = Im(Π
V
) = V
where the sum is over a set of coset representatives of H. This shows that Y
0
is non-zero. In
particular we have dim Y
0
= dim Y . We can now compute the scalar λ via
dim V = Tr(Π
V
) = Tr(
X
t
Π
V
tEt
−1
Π
V
) (since
X
t
tEt
−1
= I)
= Tr(
X
t
tΠ
V
EΠ
V
t
−1
) (since Π
V
commutes with the action of G)
= λ
X
t
Tr(tΠ
Y
0
t
−1
) (by Eq. 2)
= λ|G : H|dim Y
0
= λ dim Y
↑
which yields λ =
dim V
dim Y
↑
.
Finally, let |ψi be any unit vector in Y
0
. Consider the rank-1 projection |ψihψ| : Y
0
→ Y
0
. We
apply the averaging operator R
H
(see Section 5.1.2) to get R
H
(|ψihψ|) =
1
|H|
P
h∈H
h|ψihψ|h
−1
.
The space of H-invariant maps from Y
0
to Y
0
is 1-dimensional (by Schur’s Lemma) and spanned
by Π
Y
0
. Hence R
H
(|ψihψ|) is a scalar multiple of Π
Y
0
, and by taking traces we find R
H
(|ψihψ|) =
1
dim Y
Π
Y
0
.
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Using this we compute
hψ|Eψi = Tr(|ψihψ|E) = Tr(
1
|H|
X
h∈H
h|ψihψ|h
−1
E) (since E is H-invariant)
= Tr(
1
dim Y
Π
Y
0
E) (by above discussion)
=
1
dim Y
Tr(Π
Y
0
EΠ
Y
0
)
=
1
dim Y
Tr(λΠ
Y
0
) (by Eq. 3 above)
=
dim V
dim Y
↑
(since Tr(Π
Y
0
) = dim Y , and λ =
dim V
dim Y
↑
)
as needed.
Proof of Theorem 5.1, lower bound. Let Y be an irreducible constituent of V
↓
which maximizes
the quantity
dim(Y
↑
)
V
dim Y
↑
.
Let V
0
denote the G-subrepresentation (Y
↑
)
V
of Y
↑
and again let E denote the orthogonal pro-
jection onto the subspace e ⊗Y ⊂ Y
↑
. Then by Proposition 5.2 there exists a unit vector |ψi ∈ V
0
such that
hψ|E|ψi =
dim V
0
dim Y
↑
=
dim Y
↑
V
dim Y
↑
.
Now consider the oracle problem given by (G, V
0
, π
0
, f) (i.e. the coset identification problem
where the oracle is represented on V
0
rather than V ). Let Π
V
0
denote the G-invariant orthogonal
projection onto V
0
. We define a single-query algorithm for (G, V
0
, π
0
, f) using no ancilla, the
input state |ψi, and projective measurement {tΠ
V
0
EΠ
V
0
t
−1
}
t
where t ranges over a set of coset
representatives for H (so measuring outcome t uniquely determines a coset of H). The measurement
is used to distinguish the density operators {ρ
t
= tρt
−1
} where ρ =
1
|H|
P
h∈H
h|ψihψ|h
−1
. Note
that the support of ρ is contained in V
0
, since |ψi ∈ V
0
and V
0
is G-invariant. Therefore ρΠ
V
0
=
Π
V
0
ρ = ρ. Using this, we compute the success probability as
P
succ
=
1
|G : H|
X
t
Tr(ρ
t
tΠ
V
0
Et
−1
Π
V
0
)
=
1
|G : H|
X
t
Tr(ρ
t
tΠ
V
0
EΠ
V
0
t
−1
) (since Π
V
0
is G-equivariant, so commutes with t
−1
)
=
1
|G : H|
X
t
Tr(ρΠ
V
0
EΠ
V
0
) (since the trace is cyclic, and t
−1
ρ
t
t = ρ)
= Tr(ρE) (since the trace is cyclic, and ρΠ
V
0
= Π
V
0
ρ = ρ)
= hψ|E|ψi =
dim Y
↑
V
dim Y
↑
.
This shows that there is an algorithm for (G, V
0
, π
0
, f) which succeeds with probability
dim Y
↑
V
dim Y
↑
.
Since V
0
= (Y
↑
)
V
only contains irreps which are also contained in V , Lemma 2.4 implies there is
also an algorithm for (G, V, π, f) which succeeds with the same probability.
Remark. In applying Lemma 2.4 to produce an algorithm for (G, V, π, f), one may have to
introduce an ancilla register, to ensure that irreps appear with sufficiently large multiplicity to
allow an embedding of V
0
into the workspace.
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5.2 The upper bound
In this section we prove the upper bound of Theorem 5.1 using a minimum-error quantum state
discrimination approach [EMV06]. Before explaining the strategy to obtain the bound, we review
the set-up. We fix an instance of coset identification (G, V, π, f : G → X). The subgroup H is
the preimage of f (e), and the elements of X may be identified with the left cosets of H. A single-
query algorithm uses an initial state |ψi ∈ V and feeds it to the oracle, which is a hidden element
a sampled uniformly from G. Afterwards, a measurement {E
x
}
x∈X
is applied with the goal of
recovering f(a). With a choice of initial state fixed, the task of finding an optimal measurement
{E
x
} amounts to finding an optimal measurement to discriminate the mixed states {ρ
x
}
x∈X
, where
ρ
x
=
|X|
|G|
X
g∈G
f(g)=x
g|ψihψ|g
−1
.
Indeed, the success probability of the algorithm is equal to the probability that the measurement
{E
x
} successfully discriminates the mixed states {ρ
x
}, namely
P
succ
=
1
|X|
X
x∈X
Tr(E
x
ρ
x
).
We will prove that this success probability is bounded above by the quantity given in Theorem
5.1, which involves induced representations. We now provide an outline of the proof to give an
indication of how induced representations enter the picture.
To take advantage of symmetry in the problem, note that the density matrices {ρ
x
} always
satisfy
ρ
g·x
= gρ
x
g
−1
.
We say a set of operators with this symmetry is orbital (a precise definition is given below). We
first argue that any optimal measurement to distinguish an orbital set of density matrices can
be modified to produce another optimal measurement which is itself orbital (Lemma 5.3). Next
we aim to simplify the problem further by showing that any orbital POVM can be replaced by
a measurement which is both orbital and projective. To do so requires embedding the original
CG-module V into a larger one W by adding an ancilla register. This is the content of Lemma
5.5, which is a “symmetric” version of the usual result that any POVM can be simulated using
projective measurements and ancilla registers. As a result of this lemma we may make the following
assumptions about an optimal single query algorithm, which uses the larger Hilbert space W :
1. The measurement operators {E
x
}
x∈X
are projective and orbital.
2. The initial state |ψi belongs to a G-invariant subspace of W which is isomorphic to V .
The existence of a projective orbital measurement implied by (1) is a strong condition on the
structure of W : using the completeness relation, W can be written as the direct sum of the images
of the measurement operators {E
x
}. The subspace Y which is the image of E
x
0
is left invariant
by H, and the other subspaces are obtained through translation by a coset representative. This
realizes W as the induced representation Y
↑
. Finally, in this restricted setting (incorporating the
assumption (2) that |ψi ∈ V ) we are able to bound the success probability by decomposing Y into
irreducible H-subrepresentations, and then applying a critical inequality (Proposition 5.7) that
covers the situation when Y is irreducible. We now give the details.
With a given unitary representation V of G and a fixed G-set X understood we say a set of
operators {A
x
}
x∈X
(on V ) is orbital if gA
x
g
−1
= A
g·x
for all x and g. The density matrices for a
single query algorithm for coset identification form an orbital set.
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Lemma 5.3. Suppose {ρ
x
}
x∈X
is an orbital set of density matrices. Then there exists an optimal
measurement to distinguish the states {ρ
x
} which is orbital.
Proof. Eldar, Megretski, Verghese give the proof when X = G with the action of left multiplication
([EMV04], Section 4.3) and it works in this setting as well. We give the proof for the reader’s
convenience. Suppose {E
x
}
x∈X
is an optimal measurement. Then we define new measurement
operators {
b
E
x
}
x
by
b
E
x
=
1
|G|
X
g∈G
gE
g
−1
·x
g
−1
.
We claim that {
b
E
x
} is an orbital POVM which discriminates the states {ρ
x
} with the same success
probability as {E
x
}. Each operator
b
E
x
is a nonnegative combination of positive semi-definite
operators, hence is positive semi-definite. They satisfy the completeness relation:
X
x∈X
b
E
x
=
1
|G|
X
x∈X
g∈G
gE
g
−1
·x
g
−1
=
1
|G|
X
g∈G
I = I.
The completness relation for {E
x
} is used in the second equality. We check that the POVM {
b
E
x
}
is orbital:
h
b
E
x
h
−1
=
1
|G|
X
g∈G
hgE
g
−1
·x
g
−1
h
−1
=
1
|G|
X
k∈G
kE
k
−1
h·x
k
−1
=
b
E
h·x
.
To complete the proof it suffices to show that the new measurement discriminates the states {ρ
x
}
with the same probability as the original measurement. Indeed, we have
1
|X|
X
x∈X
Tr(
b
E
x
ρ
x
) =
1
|X||G|
X
x∈X
X
g∈G
Tr(gE
g
−1
·x
g
−1
ρ
x
) =
1
|X||G|
X
g∈G
X
x∈X
Tr(E
g
−1
·x
ρ
g
−1
·x
) =
=
1
|G|
X
g∈G
1
|X|
X
y∈X
Tr(E
y
ρ
y
)
=
1
|X|
X
y∈X
Tr(E
y
ρ
y
).
The second equality follows from the orbital assumption ρ
g
−1
·x
= g
−1
ρ
x
g, and the other steps are
index substitutions.
For the next result we use the following fact (cf. [NC11], Exercise 2.67):
Lemma 5.4. Suppose V is a unitary representation of G, W a subrepresentation, and C : W → V
a CG-module map which preserves inner products. Then C can be extended to V , meaning there
is a unitary CG-module isomorphism U : V → V such that U coincides with C on W .
Proof. Let Y denote the orthogonal complement of W and Y
0
the orthogonal complement of C(W ).
Since C preserves inner products, it is injective, so C(W )
∼
=
W as CG-modules. Hence Y
∼
=
Y
0
as CG-modules, and there exists an inner product preserving isomorphism D : Y → Y
0
. Now the
desired unitary operator U is given by
U(x) =
(
C(x) x ∈ W
D(x) x ∈ Y.
The following result is an equivariant version of the argument given by Chuang and Nielsen to
show that arbitrary measurement operators can be simulated using projective measurements and
ancilla spaces (see [NC11], Section 2.2.8).
Lemma 5.5. Suppose {E
x
} is an orbital POVM on the space V . Then there exists a unitary
representation W and an inner product preserving CG-module embedding
ι : V → W
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together with a projective orbital measurement {E
x
} on W such that for any state |ψi, the mea-
surement statistics by measuring |ψi with {E
x
} are identical to those given by the state ι|ψi and
measurement {E
x
}.
Proof. Let W be the space V ⊗CG and fix a basepoint x
0
of the G-set X.Let M
x
= (E
x
)
1/2
be the
non-negative square root of E
x
. The uniqueness of square roots implies that the set M = {M
x
}
is orbital. In addition, these constitute a set of measurement operators for the POVM, meaning
M
∗
x
M
x
= E
x
.
Now let C
M
be the controlled-M operator acting on W via
C
M
|ψ, gi =
p
|X|M
g·x
0
|ψi ⊗ |gi.
Note that C
M
is a CG-module endomorphism of W = V ⊗ CG, since
C
M
(h · |ψ, gi) = C
M
|hψ, hgi =
p
|X|M
hg·x
0
h|ψi ⊗ |hgi
=
p
|X|hgM
x
0
g
−1
|ψi ⊗ |hgi = h ·C
M
|ψ, gi.
For the third equality we used the fact that M is orbital, i.e. M
g·x
0
= gM
x
0
g
−1
. Now C
M
is not
necessarily invertible, but we claim that C
M
preserves inner products on the subspace V ⊗ |ηi,
where |ηi =
1
√
|G|
P
g∈G
|gi is the equal superposition vector in CG:
hC
M
(|ψ, ηi) | C
M
|φ, ηi =
1
|G|
X
g,h∈G
hC
M
(|ψ, gi) | C
M
|φ, hi (by the def. of |ηi)
=
|X|
|G|
X
g,h∈G
hM
g·x
0
|ψi ⊗ |gi | M
h·x
0
|φi ⊗ |hii (by the def. of C
M
)
=
|X|
|G|
X
g∈G
hψ|E
g·x
0
|φi (since hg|hi = δ
gh
and M
∗
x
M
x
= E
x
)
=
|X|
|G|
X
h∈H
X
x∈X
hψ|E
x
|φi (by writing g = th where t · x
0
= x)
=
|X|
|G|
X
h∈H
hψ|φi = hψ|φi (by the completeness relation and |H| = |G|/|X|).
Therefore, by the previous lemma, there exists a unitary CG-module endomorphism U which
restricts to C
M
on V ⊗|ηi. We are ready to define the embedding ι and measurement {E
x
} that
satisfy the claim of the theorem.
We take ι to be the inclusion of V as V ⊗ |ηi:
ι|ψi = |ψi ⊗ |ηi.
Clearly ι is an inner product preserving CG-module embedding. We define the projective mea-
surement {E
x
} by
E
x
= U
−1
X
g:g·x
0
=x
I ⊗ |gihg|
!
U.
Here I denotes the identity on V . The operators {E
x
} constitute a projective measurement, and
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we check that they form an orbital set. Let h ∈ G. Then
hE
x
h
−1
= U
−1
h
X
g:g·x
0
=x
I ⊗ |gihg|
!
h
−1
U (since U is a CG-module map)
= U
−1
X
g:g·x
0
=x
I ⊗ |hgihhg|
!
U (since h acts by π
V
(h) ⊗ h on W = V ⊗ CG)
= U
−1
X
k:k·x
0
=h
I ⊗ |kihk|
!
U = E
h·x
.
Now suppose ι|ψi = |ψi ⊗ |ηi is measured with the projective measurement {E
x
}. Then the
probability of reading outcome x is
hψ, η|E
x
|ψ, ηi = hψ, η|U
−1
X
g:g·x
0
=x
I ⊗ |gihg|
!
U|ψ, ηi
=
C
M
(|ψ, ηi)|
X
g:g·x
0
=x
I ⊗ |gihg|
!
C
M
|ψ, η
=
|X|
|G|
X
h∈G
M
h·x
0
|ψi ⊗ |hi
!
|
X
g:g·x
0
=x
I ⊗ |gihg|
!
X
h
0
∈G
M
h
0
·x
0
|ψi ⊗ |h
0
i
!
=
|X|
|G|
X
g:g·x
0
=x
hM
g·x
0
ψ|M
g·x
0
ψi =
|X|
|G|
X
g:g·x
0
=x
hM
x
ψ|M
x
ψi
= hψ|E
x
|ψi.
The first three equalities are definitions, the fourth expands the multiplication, the fifth is nota-
tional and the last follows since the number of g for which g · x
0
= x is equal to |G|/|X| for all
x ∈ X (since X is a transitive G-set). This proves the lemma.
As a result of the lemma, any orbital measurement to distinguish orbital states in a CG-module
Y can be simulated by a projective orbital measurement in a larger CG-module W . The next
lemma explains that the existence of a projective orbital measurement implies a decomposition of
the Hilbert space W that realizes W as a representation induced from H.
Lemma 5.6. Suppose {E
x
}
x∈X
is a projective orbital measurement on a CG-module W . Let W
x
denote the image of E
x
. Then W
f(e)
is an H-representation and W
∼
=
W
↑
f(e)
.
Proof. If {E
x
} is an orbital measurement then hE
f(e)
h
−1
= E
h·f(e)
= E
f(e)
for all h ∈ H, i.e.
E
f(e)
is a CH-module homomorphism. Hence the image of E
f(e)
is invariant under H.
Since the set {E
x
}
x
constitutes a measurement, W =
L
x
W
x
. Furthermore, since E
g·f(e)
=
gE
f(e)
g
−1
, we have W
g·f(e)
= gW
f(e)
. Hence W =
L
t
tW
f(e)
where the sum is over a set of
left coset representatives for H. By the characterization of induced representations discussed in
Section 5, this shows W
∼
=
W
↑
x
0
.
The lemmas above show that as long as we are willing to embed our original representation V
into a larger representation W , we may assume that W is induced from some representation Y of H
and that the measurement operators are projections corresponding to the direct sum decomposition
of W as an induced representation. In other words, the measurement operator corresponding to
outcome x ∈ X is projection onto t ⊗ Y where t is any element such that t · f(e) = x. The next
lemma is the final key to unlocking the upper bound.
Proposition 5.7. Suppose Y is an irreducible unitary representation of H (a subgroup of G).
Let V be a G-subrepresentation of Y
↑
. Let E denote orthogonal projection onto the subspace
e ⊗ Y ⊂ Y
↑
. Then for any unit vector ψ ∈ V we have
hψ|E|ψi ≤
dim V
dim Y
↑
.
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Proof. Consider the action of H on Y
↑
and let W denote the Y -isotypic component of Y
↑
. Then
W
∼
=
Y ⊗ C
m
as H-representations where m is the multiplicity of the irrep Y in Y
↑↓
. Since E is
an H-invariant projection with image isomorphic to Y , we may assume that ψ belongs to W in
addition to V . (Indeed, the support of E is contained in W , so E = Π
W
E = EΠ
W
, which implies
hψ|E|ψi = hΠ
W
ψ|EΠ
W
|ψi.) Fix an orthonormal basis {y
1
, . . . , y
d
} of Y so that we may write
|ψi =
d
X
i=1
λ
i
|y
i
, u
i
i
where the u
i
’s are unit vectors in C
m
and λ
i
≥ 0 with
P
i
λ
2
i
= 1. We apply the averaging operator
R
H
of Section 5.1.2 to the projection |ψihψ|. By Equation (1) of Section 5.1.2 we have
R
H
(|ψihψ|) =
1
dim Y
d
X
i=1
λ
2
i
(I ⊗ |u
i
ihu
i
|) .
In particular R
H
(|ψihψ|) ≤
1
dim Y
Π
W
. Note that since |ψi ∈ V ∩W , the support of R
H
(|ψihψ|) is
also contained in the H-submodule V ∩ W . Hence we deduce the stronger inequality
R
H
(|ψihψ|) ≤
1
dim Y
Π
V ∩W
.
Now we may estimate hψ|Eψi:
hψ|Eψi = Tr(E|ψihψ|) = Tr(R
H
(E|ψihψ|)) (since R
H
preserves traces)
= Tr(ER
H
(|ψihψ|) (since E is H-invariant)
≤
1
dim Y
Tr(EΠ
V ∩W
)
≤
1
dim Y
Tr(EΠ
V
).
Here Tr(EΠ
V
) can be computed by averaging over a set of coset representatives for H:
Tr(EΠ
V
) =
1
|G : H|
X
t
Tr(tEΠ
V
t
−1
)
Using that Π
V
commutes with the action of G and that
P
t
tEt
−1
= I we have
Tr(EΠ
V
) =
1
|G : H|
X
t
Tr(tEt
−1
Π
V
) =
=
1
|G : H|
Tr(Π
V
) =
dim V
|G : H|
.
Therefore
hψ|Eψi ≤
dim V
dim Y |G : H|
=
dim V
dim Y
↑
.
We are ready to prove the upper bound in Theorem 5.1.
Proof of Theorem 5.1, upper bound. Let (G, V, π, f) specify an instance of coset identification and
let H denote the stabilizer of a chosen point x
0
∈ X (recall that the codomain of f is a transitive
G-set X). Suppose an optimal single-query algorithm is given by an input state |ψi ∈ V (again
we may assume there is no workspace by absorbing it into V ) and POVM {
b
E
x
}. By Lemmas 5.5
and 5.6, there is a (not necessarily irreducible) representation Y of H and CG-submodule of Y
↑
isomorphic to V (which we identify with V ) such that the success probability of our algorithm is
equal to the success probability of an algorithm using input state |ψi ∈ V ⊂ Y
↑
and the projective
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measurement {tEt
−1
}
t
where E denotes orthogonal projection onto e ⊗ Y and t ranges over a set
of coset representatives for H.
Now decompose Y into irreducible H-invariant orthogonal subspaces:
Y = Y
1
⊕ ··· ⊕ Y
r
.
Then Y
↑
∼
=
L
i
Y
↑
i
as CG-modules. Let Π
i
denote orthogonal projection onto Y
↑
i
. Then |ψi can
be decomposed as a combination of orthogonal unit vectors
|ψi = λ
1
|ψ
1
i + ··· + λ
r
|ψ
r
i
such that each |ψ
i
i belongs to Y
↑
i
. Even more is true: since λ
i
|ψ
i
i = Π
i
|ψi and Π
i
is a CG-module
map, we know |ψ
i
i ∈ (Y
↑
i
)
V
.
Note also that E decomposes as E = E
1
+ ··· + E
r
where E
i
is orthogonal projection onto
e ⊗ Y
i
.
We are ready to bound the success probability of the algorithm. Recall that we are
using the measurement {tEt
−1
}
t
to distinguish the density operators {tρt
−1
}
t
where ρ =
1
|H|
P
h∈H
h|ψihψ|h
−1
. Then
P
succ
=
1
|G : H|
X
t
Tr((tρt
−1
)tEt
−1
) = hψ|E|ψi.
Now using the decomposition of |ψi we have
hψ|E|ψi =
r
X
i=1
|λ
i
|
2
hψ
i
|E
i
|ψ
i
i.
Now by Proposition 5.7 we have, for all i,
hψ
i
|E
i
|ψ
i
i ≤
dim(Y
↑
i
)
V
dim Y
↑
i
.
Therefore
P
succ
≤
X
i
|λ
i
|
2
dim(Y
↑
i
)
V
dim Y
↑
i
≤ max
Y ∈Irr(H)
dim Y
↑
V
dim Y
↑
.
5.3 Query complexity
We now know the success probability of an optimal single-query algorithm solving coset identi-
fication. As in Section 4, we combine this with the fact that an optimal t-query algorithm with
access to the representation V is the same as an optimal 1-query algorithm to V
⊗t
to determine
the optimal success probability for t-query algorithms:
Corollary 5.8. Let (G, V, π, f) describe a case of coset identification. Then an optimal t-query
algorithm succeeds with probability
P
opt
= max
Y ∈Irr(H)
dim Y
↑
V
⊗t
dim Y
↑
.
A straightforward consequence is the following:
Theorem 5.9. Let (G, V, π, f) describe a case of coset identification. Then the zero-error quantum
query complexity of the problem is the minimum t for which there exists some Y ∈ Irr(H) such
that every irrep of G appearing in Y
↑
also appears in V
⊗t
.
The bounded error quantum query complexity is the minimum t for which
max
Y ∈Irr(H)
dim(Y
↑
)
V
⊗t
dim Y
↑
≥ 2/3.
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6 New examples of coset identification
6.1 Identifying the coset of the Klein 4 group
Here we present an easy demonstration of the machinery of the previous section. Con-
sider the symmetric group on 4 letters G = S
4
with normal subgroup the Klein 4-group
H = {e, (12)(34), (13)(24), (14)(23)}. Given access to the defining permutation representation V
of S
4
we would like to identify which coset of H our permutation belongs to. Classically this
requires 2 queries. To determine the quantum complexity we need to know the characters of V
and S
4
. Of course V is isomorphic to Z
2
× Z
2
(say, using the generators (12)(34) and (13)(24))
and has 4 characters labelled ψ
α,β
with α, β ∈ {0, 1}. The group S
4
has 5 characters parametrized
by partitions of 4, denoted χ
[4]
, χ
[3,1]
, χ
[2
2
]
, χ
[2,1
2
]
and χ
[1
4
]
. The restriction/induction rules are
conveniently described in a Bratteli diagram (Figure 4).
ψ
0,0
ψ
0,1
ψ
1,0
ψ
1,1
χ
[4]
χ
[3,1]
χ
[2
2
]
χ
[2,1
2
]
χ
[1
4
]
Figure 4: Restriction/induction rules for H < S
4
. The irreps appearing in V are circled.
The diagram indicates, for instance, that χ
↓
[2,1
2
]
= ψ
0,1
+ψ
1,0
+ψ
1,1
and ψ
↑
0,0
= χ
[4]
+2χ
[2
2
]
+χ
[1
4
]
.
Finally, we are given access to the defining permutation representation of S
4
which decomposes as
V = χ
[4]
+ χ
[3,1]
.
To find the optimal success probability of a single-query algorithm to determine which coset
of H a permutation belongs to, we examine the irreps of H appearing in V . From the diagram
we see that every irrep of H appears in V , so we look at each one. First consider the trivial
representation ψ
0,0
. The only irrep of S
4
that appears in both V and ψ
↑
0,0
is χ
[4]
, which contributes
a one dimensional subspace to the 6 dimensional ψ
↑
0,0
. Therefore using the irrep ψ
0,0
gives a success
probability of 1/6. Now consider ψ
0,1
. In this case only χ
[3,1]
appears in both V and ψ
↑
0,1
, and it
contributes 3 dimensions to the 6 dimensional ψ
↑
0,1
. Therefore the success probability using this
irrep is 3/6 = 1/2. The other characters ψ
1,0
and ψ
1,1
give the same ratio so the optimal success
probability of a single-query quantum algorithm is 1/2 (note a single-query classical algorithm can
do no better than probability 1/6).
That the optimal 2-query success probability is 1 can be verified using the fact that V
⊗2
contains a copy of every irrep of S
4
except the sign representation, and so using any of the irreps
ψ
0,1
, ψ
1,0
, ψ
1,1
we can achieve probability 1.
6.2 An action of the Heisenberg Group
We now consider a natural action of the Heisenberg group over a finite field for which the oracle
identification problem achieves a significant quantum speedup over the best classical algorithm.
For this action, we also show that a single query suffices to solve the coset identification problem,
where the chosen subgroup H is the center of the group.
Specifically, let p be prime and let n be a positive integer. Let G = G(p, n) denote the
Heisenberg group of all (n + 2)-by-(n + 2) matrices with entries in Z
p
, 1’s on the main diagonal
and whose only other nonzero entries are in the first row and last column. Such matrices are in
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correspondence with triples (x, y, z), with x, y ∈ Z
n
p
and z ∈ Z
p
, where (1, x, z) is the first row of
the matrix and (z, y, 1) is the last column of the matrix. Then G(p, n) is a p-group of order p
2n+1
.
We consider the usual action of G(p, n) on the set X = Z
n+2
p
, considered as column vectors, by
matrix-vector multiplication. The corresponding classical oracle identification problem turns out
to have complexity b(G) = n + 1. To see this note that y and z can be determined by the single
query (0, . . . , 0, 1). Further queries give affine conditions on x, and it requires at least n of these
to determine the value of x.
In contrast to the n + 1 queries needed to solve this question classically, we now show that
a single quantum query suffices to solve the problem with high probability, and that two queries
suffice to solve the problem with certainty.
Theorem 6.1. Let G(p, n) denote the Heisenberg group defined above acting by multiplication on
the set of column vectors X = Z
n+2
p
. Then an optimal single query quantum algorithm solves the
oracle identification problem with probability
P
opt
= 1 −
1
p
+
2
p
n+1
−
1
p
2n+1
.
Furthermore, two queries suffices to solve the oracle identification problem with probability 1.
We will prove this theorem shortly. Before doing so, let us consider a related coset identification
problem. Let H < G(p, n) be the subgroup in which x = y = 0. Then H is a subgroup of order
p, and in fact H is the center of G(p, n). The coset identification problem with respect to this
subgroup H asks us to determine the values of x and y. In the classical case, n + 1 queries are
again required. However this time, a single quantum query solves the coset identification problem
with certainty.
Theorem 6.2. Let G = G(p, n) denote the Heisenberg group acting by multiplication on the set of
column vectors X = Z
n+2
p
. Let H be center of G, the set of all matrices in G for which x = y = 0.
Then the coset identification problem can be solved with a single quantum query with probability 1.
In order to prove these theorems, we must understand the representation theory of G = G(p, n),
which we now describe briefly (for a concise and elegant review, see the letter by M. Isaacs to P.
Diaconis published in the appendix to [Dia]). The group G has p
2n
one-dimensional irreducible
representations and p − 1 irreducible representations of dimension p
n
. The one-dimensional rep-
resentations will be denoted χ
α,β
, indexed by tuples α, β ∈ Z
n
p
. We identify these representations
with their characters which are given by the formula
χ
α,β
(x, y, z) = ω
α·x+β·y
,
with ω denoting a primitive p-th root of unity.
The p
n
dimensional representations denoted ρ
c
, with c ∈ Z
p
, c 6= 0 are described as follows.
Let U be the vector space of all complex-valued functions on (Z
p
)
n
. Fix c ∈ Z
p
with c 6= 0. Then
there is an irreducible representation ρ
c
of G on U given by
[ρ
c
(x, y, z)f](w) = ω
c(y·w+z)
f(w + x).
The character of this representation is given by
θ
c
(x, y, z) =
(
p
n
ω
cz
if x = y = 0,
0 otherwise.
In order to understand the query complexity of the oracle identification problem we must
decompose the representation V = C
X
into irreducible representations. Since this representation
comes from a permutation representation of G, each character value χ
V
(x, y, z) is simply the
number of fixed points of the matrix A = (x, y, z). This number of fixed points is determined
by the rank of the matrix A
0
= A − I. If (x, y, z) = 0, then A
0
has rank 0, and if x and y are
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both nonzero, then A
0
has rank 2. In all other cases A
0
has rank 1. We thus obtain the following
character values of our given permutation representation V :
χ
V
(x, y, z) =
p
n+2
if (x, y, z) = (0, 0, 0)
p
n
if x 6= 0 and y 6= 0
p
n+1
otherwise.
To find the number of copies of the trivial representation χ
0,0
appearing in χ
V
, we simply
average these values and obtain hχ
V
, χ
0,0
i = p
n
+ 2(p − 1).
Now let φ be any nontrivial irreducible character of G. We compute the number hχ
V
, φi of
copies of φ appearing in χ
V
as follows
hχ
V
, φi =
1
|G|
X
(x,y,z)∈G
χ
V
(x, y, z)φ(x, y, z) =
1
|G|
X
(x,y,z)∈G
(χ
V
(x, y, z) − p
n
)φ(x, y, z)
=
1
|G|
X
(x,y,z)
0
(p
n+1
− p
n
)φ(x, y, z) + (p
n+2
− p
n
)φ(0, 0, 0)
=
p − 1
p
n+1
X
(x,y,z)
0
φ(x, y, z) + (p + 1)φ(0, 0, 0)
where (x, y, z)
0
indicates a sum over those (x, y, z) such that x = 0 or y = 0, but (x, y, z) 6= (0, 0, 0).
In the first line, we used the fact that if φ is nontrivial then 0 = hφ, i = 1/|G|
P
(x,y,z)∈G
φ(x, y, z).
Taking φ = θ
c
in this formula, we conclude that V contains p−1 copies of ρ
c
. Taking φ = χ
α,β
,
we get
hχ
V
, χ
α,β
i =
(
p − 1 if α = 0 or β = 0, but not both
0 if α 6= 0 and β 6= 0.
We conclude that our V contains copies of all irreducible representations of G except the χ
α,β
for which both α and β are nonzero. The optimal single-query quantum success probability is thus
given by
P
opt
=
1
|G|
|G| −
X
α,β6=0
1
= 1 −
1
p
+
2
p
n+1
−
1
p
2n+1
,
as claimed.
If two queries are allowed, we have access to the representation V ⊗ V . Noting that χ
α,β
=
χ
α,0
⊗ χ
0,β
, it follows that V ⊗ V contains every irreducible representation of G. Hence, there is
a probability 1 algorithm with two quantum queries.
Finally, we turn our attention to the coset identification problem for the subgroup H =
{(0, 0, z)|z ∈ Z
p
}. To see that there is a probability one algorithm, note that any of the non-
trivial characters of H induces up to p
n
times one of the ρ
c
. Since ρ
c
is contained in V , it follows
that the coset identification problem can be solved with one query.
6.3 Guessing the sign of a permutation
Suppose there is an unknown permutation g ∈ G = S
n
for some n ≥ 2. We wish to learn the sign
of g using queries to the standard action of S
n
on {1, ..., n}. This is an instance of the hidden coset
problem where H = A
n
. Classically, n − 1 queries are necessary to determine the sign of g. In
fact, any fewer queries and we do not learn anything about the sign. Quantumly, we have
Theorem 6.3. Let n ≥ 2 and consider the standard action of S
n
on {1, . . . , n}. Consider the
hidden coset problem for the subgroup H = A
n
. That is we wish to determine the sign of a hidden
permutation. For exact learning, t = b
n
2
c quantum queries suffice. With any smaller number of
quantum queries, one cannot do any better than random guessing (p = 1/2.)
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Proof. For facts and notation about representations of S
n
and A
n
, we refer the reader to the proofs
of Theorems 4.3 and 4.4.
Let V be the defining representation of S
n
, and suppose we use t queries so that we have
access to V
0
= V
⊗t
. Suppose λ is a non-self-conjugate partition such that V
0
contains V
λ
. Letting
Y = W
λ
, we see that Y
↑
consists of one copy of V
λ
and one copy of V
λ
∗
. Hence the quotient of
dimensions
dim (Y
↑
)
V
0
dim Y
↑
equals 1 if V
0
contains both V
λ
and V
λ
∗
and
1
2
if V
0
contains V
λ
but not V
λ
∗
. Now consider a
self-conjugate partition λ contained in V
0
. In this case, if we take Y = W
+
λ
, then Y
↑
is V
λ
. Hence
in this case the quotient of dimensions is 1.
We thus wish to find the smallest t such that V
⊗t
contains both V
λ
and V
λ
∗
for some partition
λ (including the possibility that λ is self-conjugate). For such t, we will have a t-query probability
1 algorithm and for fewer queries we cannot do better than probability 1/2, which is random
guessing.
For even n, the value t = n/2 produces the partition λ = (n/2 + 1, 1, . . . , 1) (with n/2 − 1 1’s)
and its conjugate λ
∗
= (n/2, 1, . . . , 1) (with n/2 1’s). For odd n, the value t =
n−1
2
produces the
self conjugate partition (
n+1
2
, 1, . . . , 1) (with
n−1
2
1’s). In either case t = b
n
2
c gives a probability 1
success, and fewer queries give success probability 1/2.
7 Previously studied examples of coset identification
Here we discuss the relation of this work to preceding work. To the authors knowledge, every
previously studied special case of the general coset identification problem uses oracles sampled
from an abelian group. Zhandry [Zha15] addresses this problem (calling it the oracle classification
problem) and provides an expression for the optimal success probability essentially identical to 5.1.
Thus our results are a non-abelian generalization of Zhandry’s work, which was a key inspiration
for the present paper. We briefly explain why Zhandry’s result is equivalent to ours and then
examine some other more specialized and well-known problems.
Coset identification for an abelian group is described by a tuple (A, V, π, f) with A abelian
and f : A → X distinct and constant on the cosets of a subgroup B. We remark that since B
is a normal subgroup it is possible to identify X with the quotient group A/B and f with the
standard homomorphism A → A/B. Hence coset identification in this instance may also be called
homomorphism evaluation. By Cor. 5.8 the optimal success probability for a t-query algorithm to
determine f(a) is
P
opt
= max
Y ∈Irr(B)
dim Y
↑
V
⊗t
dim Y
↑
.
Since B is abelian, Y is 1-dimensional and Y
↑
decomposes as |A : B| many distinct A-characters
(corresponding to the characters of A/B). Hence dim Y
↑
V
⊗t
(which by definition is the dimension
of the maximal subspace of Y
↑
containing only characters in V
⊗t
) is exactly equal to the number
of shared irreps, i.e. the cardinality of the set I(Y
↑
) ∩ I(V
⊗t
). As Y varies, these sets partition
I(V
⊗t
) into equivalence classes [χ], and by Frobenius reciprocity two characters are equivalent if
and only if their restrictions to B are identical. Hence the equation above can be restated:
Theorem 7.1. (Zhandry, ([Zha15], Theorem 4.1)) The optimal success probability of a t-query
algorithm for abelian coset identification is
P
opt
=
1
|A : B|
max
χ∈I(V
⊗t
)
|[χ]|.
Under this interpretation we’re aiming to find the largest collection of characters appearing
in V
⊗t
which have the same restriction to B. Zhandry includes several nice applications of the
previous theorem, explained in a linear algebraic framework. Below we readdress a couple of these
problems (polynomial interpolation and group summation) using character theoretic language, and
we revisit the van Dam algorithm [van98].
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7.1 Polynomial interpolation
The polynomial interpolation problem as outlined by Zhandry [Zha15] and Childs, van Dam, Hung
and Shparlinski [CvDHS16] is as follows. Let F = F
q
where q = p
r
for some prime p. Suppose
we have an unknown polynomial f(X) over F of degree less than or equal to d and we wish to
determine f using queries that provide the value f(x) for x ∈ F . That is access to f is provided
via the oracle U
f
acting on V = C
F
⊗ C
F
by
U
f
: |x, si 7→ |x, s + f(x)i.
This equation defines a representation on V of the group G of all polynomials of degree less than
or equal to d under addition.
We would like to see which of the characters of G appear in this representation. Let ω be a
primitive p-th root of unity and let Tr denote the trace map from F
q
to F
p
. The characters of the
additive group F are given by χ
y
with y ∈ F defined by
χ
y
(x) = ω
Tr(yx)
.
For y ∈ F , define the character state
|ω
y
i =
X
s∈F
χ
y
(−s)|si.
It is easy to see that
U
f
|x, ω
y
i = χ
y
(f(x))|x, ω
y
i.
Thus if we let V
x,y
denote the 1-dimensional space spanned by |x, ω
y
i, we have the decomposition
V =
M
V
x,y
into irreducible representations.
The characters of F
d+1
, which is isomorphic to G, are given, for a ∈ F
d+1
, by φ
a
, where
φ
a
(c) = ω
Tr(a·c)
The character of V
x,y
is φ
a
, with
a = (y, yx, yx
2
, . . . , yx
d
). (4)
To see this note that if f(x) =
P
c
i
X
i
, then
χ
y
(f(x)) = ω
Tr(yf (x))
= ω
c·(y,yx,yx
2
,...,yx
d
)
Thus the irreps that appear in V are exactly the φ
a
, where a has the form in Equation 4. Since
φ
a
⊗φ
a
0
= φ
a+a
0
, it follows that the k-fold tensor power contains those φ
b
where b can be expressed
as a k-fold sum of vectors of the form in Equation 4. This is exactly in image of the map Z
as described by Childs, van Dam, Hung and Shparlinski [CvDHS16], so we have reproved their
Theorem 1.
The computation of the optimal success probability is now reduced to an alge-
braic/combinatorial problem which is nontrivial to solve (and is achieved in [CvDHS16]). Hence
this example serves to show the limitations of our main results: they can be used to translate
questions about query complexity into purely algebraic problems which may or may not be easily
solvable. The character theoretic technique shown above could also be used to reduce the query
complexity of multivariable polynomial interpolation to a counting problem, as was achieved by
Chen, Childs and Hung [CCH18] without referring to characters. So far though, the character
based language has not led to any progress on this problem.
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7.2 Group summation problem
Fix an abelian group G. The k-element group summation problem is the task of computing the
sum f(1) + ··· + f(k) given access to an evaluation oracle hiding a function f : {1, 2, . . . k} → G.
This is an instance of coset identification. The oracles form a representation of the group of
functions Fun([k], G) = {f : {1, . . . , k} → G}, which we identify with G
k
. In the quantum version
they act on the Hilbert space V = C
k
⊗ CG via
U
f
|i, bi = |i, b + f(i)i.
We wish to determine Σ(f) :=
P
k
i=1
f(i), which is the same as determining the coset of f w.r.t.
the subgroup
H = {f : Σ(f) = 0} ≤ Fun([k], G).
The irreducible characters of Fun([k], G) are all of the form χ
1
× ··· × χ
k
: G
k
→ C, where each
χ
i
∈ Irr(G). The Hamming weight of such a character, denoted wt(χ
1
× ··· ×χ
k
), is the number
of components which are nontrivial. The characters appearing in the evaluation representation
on V = C
k
⊗ CG are exactly those with Hamming weight ≤ 1. This implies that the characters
appearing in V
⊗t
are those with Hamming weight ≤ t.
Next we consider the irreps of the subgroup H. We may H with G
k−1
via
(a
1
, . . . , a
k−1
, −(a
1
+ ··· + a
k−1
)) ↔ (a
1
, . . . , a
k−1
).
Hence irreps of H may be written as τ
1
×···× τ
k−1
where again the τ
i
are irreps of G. Using the
above equation one verifies that two irreps χ
1
×···×χ
k
and η
1
×···×η
k
have the same restriction
to H if and only if there exists ψ ∈ Irr(G) such that
χ
1
× ··· × χ
k
= ψη
1
× ··· × ψη
k
.
By Zhandry’s theorem (Thm. 7.1) the optimal success probability for a t-query algorithm is
obtained by finding the largest collection of characters in V
⊗t
which restrict to the same irrep of
H. We can describe an element of such a maximal equivalence class: the character χ
1
× ··· × χ
k
should have at least k −t trivial components (to guarantee its Hamming weight is ≤ t), then k −t
components equal to some nontrivial character ψ
1
(so then χ
1
ψ
−1
1
×···×χ
k
ψ
−1
also has Hamming
weight ≤ t), another k − t components equal to ψ
2
, and so on. For instance, we may pick
χ = × × ··· × ψ
1
× ψ
1
× ··· × ψ
N
× ψ
N
. . .
where the characters , ψ
1
, . . . , ψ
N
are distinct (but otherwise arbitrary), and each one appears at
least k −t many times. Then the equivalence class of χ has size N + 1, consisting of the characters
[χ] = {χ,
ψ
−1
1
× ψ
−1
1
··· × × × ··· × ψ
−1
1
ψ
N
× ψ
−1
1
ψ
N
× . . . ,
.
.
.
ψ
−1
N
× ψ
−1
N
× ··· × ψ
−1
N
ψ
1
× ψ
−1
N
ψ
1
× ··· × × × . . . }
The size N + 1 of this equivalence class is either |G| (if we can fit every irrep of G, which
happens iff b
k
k−t
c ≥ |G|) or b
k
k−t
c. Hence for a t-query algorithm
P
opt
=
1
|G|
min
k
k − t
, |G|
.
This is exactly Thm. 5.1 by Zhandry ([Zha15]). An efficient algorithm achieving this success
probability had previously been described (for G cyclic) by Meyer and Pommersheim [MP14].
Accepted in Quantum 2021-03-03, click title to verify. Published under CC-BY 4.0. 30
7.3 The van Dam algorithm
The van Dam learning problem [van98] is concerned with identifying a (total) Boolean function
f : {1, . . . , n} → Z
2
given access to evaluation queries. This is a special case of symmetric oracle
discrimination (see Section 4). The group of oracles is isomorphic to Z
n
2
and irreps can be again
written as a product χ
1
×···×χ
n
of characters of Z
2
. The characters appearing in the t-th tensor
power of the evaluation oracle representation are exactly those with Hamming weight ≤ t. Hence
the optimal success probability of a t-query algorithm is
P
opt
=
1
2
n
|{characters of Z
n
2
with wt ≤ t }| =
1
2
n
t
X
i=0
n
i
which reproves the optimality of van Dam’s algorithm.
Acknowledgements
We would like to thank Andrew Childs, Hanspeter Kraft, David Meyer, Marino Romero and Mark
Zhandry for helpful communications. We are grateful to the reviewers for their useful comments
which have given us direction towards future work.
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