Quantum and Classical Query Complexities for Generalized Simon’s Problem

# Quantum and Classical Query Complexities for Generalized Simon’s Problem

Zhenggang Wu Daowen Qiu Guangya Cai Yinuo Lin Yikang Zhu Institute of Computer Science Theory, School of Data and Computer Science, Sun Yat-sen University, Guangzhou 510006, China
The Guangdong Key Laboratory of Information Security Technology, Sun Yat-sen University, 510006, China
###### Abstract

Simon’s problem conceived by Simon is one of the most important examples demonstrating the faster speed of quantum computers than classical computers for solving some problems, and the optimal separation between exact quantum and classical query complexities for Simon’s Problem has been proved by Cai and Qiu. Generalized Simon’s problem is a generalization of Simon’s problem, and it can be described as: Given a Boolean function , , promised to satisfy the property that, for some subgroup , we have, for any , if and only if , where for some . The problem is to find . For the case of , it is the Simon’s problem. In this paper, we propose an exact quantum algorithm with queries, and an exact classical algorithm with queries for solving the generalized Simon’s problem. Then we show that the lower bounds on its exact quantum and classical deterministic query complexities are and respectively. Therefore, we obtain the tight exact quantum query complexity , and the classical deterministic query complexities for the generalized Simon’s problem.

###### keywords:
Quantum computing, Exact query complexity, Generalized Simon’s problem, Dimensional reduction

## 1 Introduction

The quantum query models are the quantum analog to the classical Boolean decision tree models, and are at least as powerful as the classical decision tree models BW02 (). A quantum query algorithm can be described by the implementation procedure of a quantum query model as: it starts with a fixed starting state of a Hilbert and will perform the sequence of operations , where ’s are unitary operators that do not depend on the input but the query does. This leads to the final state . The result is obtained by measuring the final state .

A quantum query algorithm exactly computes a Boolean function if its output equals with probability 1, for all input . computes with bounded-error if its output equals with probability at least , for all input . The exact quantum query complexity denoted by is the minimum number of queries used by any quantum algorithm which computes exactly for all input .

Simon’s problem conceived by Simon in 1994 Simon1994 () is in the model of decision tree complexity or query complexity and it is a famous computational problem that achieves exponential separation in query complexities. This problem can be described as: Given a Boolean function , , promised to satisfy the property that, for some , we have, for any , if and only if . The problem is to find .

In the bounded-error setting, Simon gave an elegant quantum algorithm which solves the problem with queries and the physical realization has demonstrated its efficiency Tame2014 (). The lower bound was proved in Koiran2007 () by using polynomial method Beals2001 (). On the other hand, the classical probabilistic query complexity for this problem is Wolf2013 (), which shows that the versus separation is an optimal one.

For the exact query complexities of Simon’s problem, Brassard and Høyer Brassard1997 () first gave an exact quantum algorithm solving the problem with queries. However, their algorithm is quite complicated. Then Mihara and Sung Mihara2003 () proposed a simpler exact algorithm with queries as well, but they did not show the construction of their oracles. Recently, Cai and Qiu CQ2018 () gave a new exact quantum algorithm for solving Simon’s problem also with queries, and it is simpler and more concrete than the previous exact quantum algorithms for Simon’s problem. In particular, Cai and Qiu CQ2018 () designed a classical deterministic algorithm for solving Simon’s problem with queries, and the classical deterministic query complexity for Simon’s problem was thus obtained. Therefore the optimal separation in the exact query complexities for Simon’s problem is versus .

It is worth mentioning that Simon’s problem over general group and Simon’s problem for linear functions have been studied Alagic2008 (); Apeldoorn2018 (). Alagic et al.Alagic2008 () investigated the Simon’s Problem over general groups , with the promise being changed, and designed a quantum algorithm with time complexity . Apeldoorn et al. Apeldoorn2018 () investigated the Simon’s problem for linear functions over , where is a prime power and is a finite field with elements, and they showed the lower bound is .

Generalized Simon’s problem proposed in KLM2007 () is a generalization of Simon’s problem, and it can be described as: Given a Boolean function , , promised to satisfy the property that, for some subgroup , we have, for any , if and only if , where for some . The problem is to find . For the case of , it is the Simon’s problem.

With bounded-error computing the generalized Simon’s problem, the authors in KLM2007 () gave an upper bound on quantum query complexity with successful probability at least . However, we still do not know the exact quantum query complexity and classical deterministic query complexity for the generalized Simon’s problem, and the optimal separation in exact quantum and classical deterministic query complexity for this problem needs to be clarified. So, in this paper, we propose an exact quantum algorithm with queries, and a classical deterministic algorithm with queries for solving the generalized Simon’s problem. Then we show that the lower bounds on its exact quantum and classical deterministic query complexities are and respectively. Therefore, we obtain the tight exact quantum query complexity , and the classical deterministic query complexities for the generalized Simon’s problem. Clearly when , it accords with the results for Simon’s problem obtained by Cai and Qiu CQ2018 ().

The remainder of the paper is organized as follows. In Section 2, we introduce a number of notations and results that will be used in the paper, and we also present the basic ideas for designing the exact quantum algorithms in the paper. Then in Section 3, we study in detail the exact quantum query complexity for the generalized Simon’s problem, and design an exact quantum algorithm with queries and show the lower bound on the exact quantum query complexity for the generalized Simon’s problem. Therefore we obtain the tight exact quantum query complexity . After that, in Section 4, we investigate the classical deterministic query complexity for the generalized Simon’s problem. In this section, we design a classical deterministic algorithm with queries and derive the lower bound on the classical deterministic complexity for the generalized Simon’s problem. Finally, conclusions are summarized in Section 5.

## 2 Preliminaries

In this section, we would present related definitions and notations, and give some properties of the generalized Simon’s problem, as well as provide the key ideas of designing an exact quantum algorithm for the generalized Simon’s problem.

### 2.1 Definitions and notations

Let with and . By , we denote the bitwise exclusive-or operation, i.e.,

.

By , we denote the inner product modulo 2 of and , i.e.,

mod 2.

Let . is the subset of defined by

.

By , we denote the cardinality of , i.e., “the number of elements of ”. The query set of , denoted by , is the subset of , satisfying

.

If is a subgroup of , denote by the dimension of .

Let be any boolean function. We use to denote the range of , to denote the domain of , to denote the codomain of , and is the support of defined by

.

We use to denote an index set, i.e., . Assume , , and we define to be an subprocedure that randomly outputs an index set , i.e., randomly takes out elements from .

### 2.2 Generalized Simon’s problem and some properties

The generalized Simon’s problem can be defined as follows, where from the promise it follows that is a subgroup of :

Given: .

Promise: For all , .

Problem: Find the set .

###### Lemma 1.

Let be a set defined in the generalized Simon’s problem. Then .

###### Proof.

By the promise , we have , and therefore . ∎

###### Lemma 2.

Let be a set defined in the generalized Simon’s problem, and . Then the is an Abel group, and is a subgroup of .

###### Proof.

satisfies the following properties:

1.

2.

3.

4.

Therefore, is an Abelian Group. By the Lemma 1, , , and clearly is also an Abelian Group. ∎

###### Definition 1.

Let , and . We call as a linearly independent set of G if and only if ,

.

###### Lemma 3.

Let be a linearly independent set of . Denote . Then is a subgroup generated by , and .

###### Proof.

From the definition of we can easily know that is s subgroup of . For any with , then . Therefore, . ∎

###### Definition 2.

Let be a subgroup of , and let be a linearly independent set of . We define the dimension of equals , if .

###### Theorem 1.

Let be defined in the generalized Simon’s problem, and . Then there exists a linearly independent set such that , and .

###### Proof.

Suppose , and is a linearly independent set of . Then, for any , .

By Lemma 3, , and . Therefore, there must exist , and is also a linearly independent set of .

By induction, we can get a linearly independent set such that , and . ∎

###### Theorem 2.

Let be defined in the generalized Simon’s problem, and . Then is a subgroup of , , and .

###### Proof.

. In addition, , and then is a subgroup of .

By Theorem 1, there exists a linearly independent set such that , and . We extend to a linearly independent set with such that .

Then we can clearly verify that .

For a better understanding, we provide some fundamental analyses based on the promise of this problem through above lemma, and a concise sketch of to illustrate the relation of mapping, where . There are exact unique images for this map, and for each element in its preimage is a set with elements. As the Figure 1 shows, the left part, representing , is a grid of , whose elements of different rows will be mapped to inequal elements in , and is a subset of with unique elements selected from different rows.

### 2.3 Dimensional reduction

Dimensional reduction, a key idea in this paper, is a practical method used in whole algorithms occurred in this paper, which uses the known results in or in to ensure next result linearly independent with previous. Brassard et al. Brassard1997 () mentioned it in their paper published in 1997, and came up with an exact quantum polynomial-time algorithm to solve Simon’s problem, but their way to analyze and practice are sophisticated to an extent. In this section, we would give relatively concise introduction to it in this section.

Suppose there exists an algorithm to randomly get a nonzero element (or ). Then we can use dimensional reduction to ensure the number of calling this algorithm only (or or the latter case), where we use the notation “(or . . . )” in this section to represent the second case.

For , let with , and denote and .

###### Remark 1.

can be divided into two parts as , since .

###### Remark 2.

. By induction, and will be two linearly independent sets of

###### Lemma 4.

We have two properties as follows:

1. ;

2. .

Now, we can draw a conclusion that the dimension of or will be reduced after we get a new or , and then we can use this trick to keep the output set to be linearly independent for designing an exact quantum or classical algorithm, or analyzing the lower bound of classical probabilistic algorithm.

### 2.4 Quantum amplitude amplification

Let us recall quantum amplitude amplification Brassard2012 ().

###### Definition 3.

Let be any quantum algorithm that uses no measurements, and let be any Boolean function. Assume that , and we call as the good state, and as the bad state, where , .

###### Lemma 5 (Brassard2012 ()).

There exists a quantum algorithm that given the initial success probability of , finds a good solution with certainty using a number of applications of and which is in in the worst case.

The complementary description of Lemma 5 is given as follows, where and are parameters dependent of :

 S0(ϕ)|x⟩={|x⟩ifx=0,eiϕ|x⟩ifx≠0,

Let . Then

,

,

where .

###### Corollary 1.

There exists a quantum algorithm that given the initial success probability of , finds a good solution with certainty using applications of and exactly both once. Let and the specific expression of the two parameters used in this algorithm is given as follows:

 {ϕ=θ+2k1π,k1∈Z,φ=θ+2k2π,k2∈Z.
###### Proof.

By Lemma 6, the chosen , satisfy Eq. (1):

 eiφ(1−eiϕ)a=((1−eiϕ)a+eiϕ), % where 0

The definition of Logarithmic Function for complex number is shown in Eq. (2):

 eW=Z⇒W=logZ=log|Z|+i(argZ+2kπ),k∈Z. (2)

Let . If , then is a pure imaginary number, and . Therefore, we get the following equations:

 |Z| =−2a2(cosφ−1)2acosφ−2a+2a2−2a2cosφ+1=1 (3) ⇒−2a2cosφ+2a2=2acosφ−2a+2a2−2a2cosφ+1 ⇒2acosφ−2a+1=0 ⇒cosφ=1−12a ⇒φ=±arccos(1−12a)+2k1π,k1∈Z.

By the denominator of being nonzero, we get the first constriction from Eq. (3):

 cosφ(2a−2a2)−2a+2a2+1≠0 ⇒ cosφ≠2a−2a2−12a−2a2=1−12a−2a2 ⇒ 1−12a≠1−12a−2a2 ⇒ a2≠0.

By the domain of defined in , we get another constriction:

 −1≤1−12a≤1 ⇒a≥14.

So, we get , with the condition . Let . Substitute into , and then , thus . ∎

## 3 Exact quantum query complexity for the generalized Simon’s problem

In this section, our purposes are to show that the lower bounds on the exact quantum query complexity is , and then to propose an exact quantum algorithm with queries for solving the generalized Simon’s problem.

### 3.1 The lower bound

Koiran Koiran2007 () gave a specific lower bound proof for Simon’s problem. They transformed Simon’s problem to another problem to distinguish between the trivial subgroup and a hiding subgroup, i.e., to determine whether or not the given is a bijection. Although the discrimination does not give the result as , the complexity of this new problem is the lower bound of Simon’s problem.

We utilize their methods but change the second property of (see Definition 7), and give a lower bound on the query complexity for the generalized Simon’s problem.

In this section, we denote by an abelian group , and denote by the set .

###### Definition 4.

Let be a partial function, and let be a total function. denotes the size of the domain of , and we define:

 Ih(f)={1if f extends h,0otherwise.

More precisely,

 Ih(f)=∏i∈domain(h),j=h(i)Δi,j(f),

where is 1 if and 0 otherwise. Then is a monomial in the variables .

###### Definition 5.

Let . We call hiding a subgroup of with order , if , , .

###### Remark 3.

For the Generalized Simon’s problem defined in Section 2.2, we have the given hiding a subgroup of with order .

###### Lemma 7 (Beals1998 ()Koiran2007 ()).

If is an algorithm of query complexity T, then there is a set of partial functions from such that, for all function , the algorithm accepts with probability

 Pn(f)=∑f∈Kαn,fIh(f),

where, for every , we have and is real number.

###### Definition 6.

An algorithm is said to distinguish the generalized Simon’s problem with bounded error , if it accepts any function hiding a subgroup of order with a probability at least and rejects every other function with a probability at least , and the query complexity is the function .

###### Remark 4.

We have to point out that algorithm only distinguishes between the subgroup of order and the other subgroup, so is the lower bound of the generalized Simon’s problem.

###### Definition 7.

Suppose is an algorithm that distinguishes the generalized Simon’s problem of error bounded by , for , and , and let be the probability that accepts when is chosen uniformly at random among the functions from to hiding a subgroup of with order . If we denote by the set of functions hiding a subgroup of order , then we have:

 Qn(D)=1|XD|∑f∈XDPn(f).

It has the following two properties.
1. For any integer , ;
2. and , hence , for some .

From the above definition, is the probability that accepts , with hiding a subgroup of of order 1, and the subgroup has only one element . As for , it represents the probability that accepts , with hiding a subgroup of of order .

We recall a useful lemma by Koiran (Koiran2007 (), Lemma 5).

###### Lemma 8 (Koiran2007 ()).

Let and be constants and let be a real polynomial with following properties:
1. for any integer , we have ;
2. for some real number , we have . Then and, more precisely,

 deg(P)≥min⎛⎜ ⎜⎝n2,log2(ξn+3c)−1log2(ξ3ξ−1)+1⎞⎟ ⎟⎠.

Now, we give a similar lemma as above Lemma 8, but change some conditions and provide a simplified proof in this section.

###### Lemma 9.

Let be a constant and let P be a real polynomial with following properties:
1. for any integer , we have ;
2. for some real number , we have .
Then

 deg(P)≥min(n−k2,n+2+log2c4).
###### Proof.

In the interest of readability, we would give the detailed proof here. Let denote the degree of . If , the proof is complete. Besides, if , it can not satisfy the second condition, and if , is a nonzero constant, so is a monotone and does not satisfy the first condition. So we assume .

The polynomial and are respectively, of degrees and , so there exists an integer such that has no real root in , and has no root whose real part is in this same interval. Then, we have two properties as follows:

1. and are constants greater than zero or constants less than zero, i.e., they does not change the sign in this interval;

2. and are monotone in this interval.

By the condition , the range of in this interval is a subset of . Then we finished the first part of this proof:

 |P′((2a+2a+1)2)| =|P′(322a)| (4) ≤max⎛⎝∣∣ ∣∣f(2a)−f(322a)2a−322a∣∣ ∣∣,∣∣ ∣∣f(2a+1)−f(322a)2a+1−322a∣∣ ∣∣⎞⎠ ≤12(a−1)max(∣∣∣f(2a)−f(322a)∣∣∣,∣∣∣f(2a+1)−f(322a)∣∣∣) ≤12(a−2).

By Eq.(4), we therefore have:

 ∣∣ ∣∣P′(322a)P′(x0)∣∣ ∣∣≤1c2a−2≤1c2n−2d. (5)

Let us write , where the s are real or complex numbers. We have the following equality:

 ∣∣ ∣∣P′(322a)P′(x0)∣∣ ∣∣=d−1∏i=1∣∣ ∣∣322a−αix0−αi∣∣ ∣∣. (6)

Let .By , then . If , then . Notice that no root of has its real part in . Suppose . We therefore have

 f(R(αi))≥14, (7)
 ∣∣ ∣∣322a−αix0−αi∣∣ ∣∣≥  ⎷(322a−R(αi))2+I2(αi)(x0−R(αi))2+I2(αi)≥min⎛⎝1,∣∣ ∣∣322a−R(αi)x0−R(αi)∣∣ ∣∣⎞⎠.

and thus by Eq.(7). We conclude from Eq.(6) that

 ∣∣ ∣∣P′(322a)P′(x0)∣∣ ∣∣≥14d−1. (8)

Taking eq.(5) into account, we finally obtain the inequality

 14d−1≤1c2n−2d. (9)

hence

 d≥n+2+log2c4.

###### Theorem 3.

If is an algorithm that solves the generalized Simon’s problem with bounded error probability and query complexity T, then ; more precisely,

 T(n)≥min(n−k4,n−k+3+log2(1−2ϵ)8).
###### Proof.

By the two properties of , an application of Lemma 9 to polynomial yields the inequality

 deg(P) ≥min⎛⎝n−k2,n+2+log22−4ϵ2k−14⎞⎠ ≥min(n−k2,n−k+3+log2(1−2ϵ)4).

Since (see, for example, Koiran2007 (), Proposition 1) and , the proof is completed. ∎

Let the bounded error in Theorem 3. Then we can get a lower bound for quantum query complexity for the generalized Simon’s problem.

###### Corollary 2.

Any exact quantum algorithm that solves the generalized Simon’s problem requires queries.

### 3.2 The upper bound

Let , and let be an index set, which is constructed recursively by Algorithm 2 with an initial condition . We use to construct the set and the quantum circuit as follows:

, ,

 f(x)={0,i∈I(l),1,i∉I(l).

is the quantum circuit using quantum amplitude amplification to remove zero state with known amplitude (see Section 2.4) which determines its construction.