Exact distributed quantum algorithm for generalized Simon’s problem

Quantum computing nielsen_quantum_2010 has been proved to have great potential in factorizing large numbers shor_polynomial-time_1997 , searching unordered database grover_fast_1996 and solving linear systems of equations HHL_2009 . However, large-scale universal quantum computers have not yet been realized due to the limitations of current physical devices. At present, quantum technology has been entered to the Noisy Intermediate-Scale Quantum (NISQ) era preskill_quantum_2018 , which makes it possible to implement quantum algorithms on middle-scale circuits.

Distributed quantum computing is a novel computing architecture, which combines quantum computing with distributed computing goos_distributed_2003 ; beals_efficient_2013 ; Qiu2017DQC ; caleffi_quantum_2018 ; avron_quantum_2021 ; Qiu22 ; Tan2022DQCSimon ; Xiao2023DQAShor ; Hao2023DDJ ; Xiao2023DQAkShor . In distributed quantum computing architecture, multiple quantum computing nodes communicate with each other and cooperate to complete computing tasks. Compared with centralized quantum computing, the size and depth of circuit can be reduced by using distributed quantum computing, which is beneficial to improve the performance of circuit against noise.

Simon’s problem is one of the most important problems in quantum computing simon_power_1997 . For solving Simon’s problem, quantum algorithms have the advantage of exponential acceleration over classical algorithms cai_optimal_2018 . Remarkably, Simon’s algorithm greatly inspired the proposal of Shor’s algorithm shor_polynomial-time_1997 . Furthermore, the generalized Simon’s problem is a natural extension of Simon’s problem, and an instance of the hidden subgroup problem nielsen_quantum_2010 ; kaye_introduction_2007 .

Tan, Xiao, and Qiu et al. Tan2022DQCSimon proposed a distributed quantum algorithm for Simon’s problem, but they left it open as to whether an exact distributed version exists. In the centralized case, Cai and Qiu cai_optimal_2018 utilized quantum amplitude amplification to address the issue of exactness for Simon’s problem. Their approach has served as inspiration for our work. In this paper, we contribute in two new ways. Firstly, we characterize the structure of the generalized Simon’s problem in distributed scenario and leverage this understanding to design a corresponding distributed quantum algorithm. Secondly, we incorporate quantum amplitude amplification BHMT02 to ensure the algorithm’s exactness.

The remainder of this paper is organized as follows. In Sec. II, we present some notations related to group theory, and recall the generalized Simon’s problem. In Sec. III, we characterize the structure of the generalized Simon’s problem in distributed scenario. Then, in Sec. IV we describe a distributed quantum algorithm for the generalized Simon’s problem and give the corresponding analytical procedure. Furthermore, in Sec. V we introduce quantum amplitude amplification technique, and with this technique, we in Sec. VI design an exact distributed quantum algorithm for the generalized Simon’s problem and prove its correctness. In addition, in Sec. VII, we compare our algorithm with other algorithms. Finally, we conclude with a summary in Sec. VIII.

In this section, we present some notations related to group theory, and recall the generalized Simon’s problem.

In the following, we describe the generalized Simon’s problem in distributed scenario and characterize its structure.

The function $f$ corresponding to the generalized Simon’s problem is divided into $2^{t}$ subfunctions $f_{w}:\{0,1\}^{n-t}\rightarrow\{0,1\}^{m}$ as follows. Let

where $u\in\{0,1\}^{n-t}$, $w\in\{0,1\}^{t}$.

Suppose there are $2^{t}$ people, each of whom has an oracle $O_{f_{w}}$ that can query all $f_{w}(u)=f(uw)$ for any $u\in\{0,1\}^{n-t}$, $w\in\{0,1\}^{t}$, where $O_{f_{w}}$ are defined as

where $u\in\{0,1\}^{n-t}$, $w\in\{0,1\}^{t}$ and $b\in\{0,1\}^{m}$.

Each person can access $2^{n-t}$ values of $f$. They need to find the hidden subgroup $S$ by querying their own oracle and communicating with each other as few times as possible.

Below, we further introduce some notations related to the function $f$ corresponding to the generalized Simon’s problem. We anticipate that the reader is already familiar with the concept of multisets.

Let $S$ be the hidden subgroup to be found, and denote the basis of $S$ as $\{s_{il}s_{ir}|s_{i}=s_{il}s_{ir},s_{i}\in\{0,1\}^{n},s_{il}\in\{0,1\}^{n-t},s_{ir}\in\{0,1\}^{t},1\leq i\leq k\}$.

Let

then $S_{l}\leq\mathbb{Z}_{2}^{n-t}$.

Denote $k_{l}=rank(S_{l})$, $0\leq k_{l}\leq\min(k,n-t)$, and the basis of $S_{l}$ as $\left\{e_{i}|e_{i}\in\{0,1\}^{n-t},1\leq i\leq k_{l}\right\}$.

Let

then $S_{r}\leq\mathbb{Z}_{2}^{t}$.

The following theorem concerning $S(u)$ is useful and important.

In the following, we begin with giving related notation, function and operators that are used in distributed quantum algorithm for the generalized Simon’s problem, i.e., Algorithm 1.

Let $[N]$ represent the set of integers $\{0,1,\cdots,2^{t}-1\}$, and let ${\rm BI}:\{0,1\}^{t}\rightarrow[N]$ be the function to convert a binary string of $t$ bits to an equal decimal integer.

The query operators $O^{\prime}_{f_{w}}$ in Algorithm 1 are defined as

where $u\in\{0,1\}^{n-t}$, $w\in\{0,1\}^{t}$, $b\in\{0,1\}^{{\rm BI}(w)}$ and $c\in\{0,1\}^{m}$.

The operator $U_{Sort}:\{0,1\}^{2^{t+1}m}\rightarrow\{0,1\}^{2^{t+1}m}$ in Algorithm 1 is defined as

where $\bigotimes\limits_{w\in\{0,1\}^{t}}\ket{f_{w}(u)}\triangleq\ket{f_{0^{t}}(u)}\ket{f_{0^{t-1}1}(u)}\cdots\ket{f_{1^{t}}(u)}$ and $b\in\{0,1\}^{2^{t}m}$.

Intuitively, the effect of $U_{Sort}$ in Algorithm 1 is to sort the values in the $2^{t}$ control registers by lexicographical order and XOR to the target register.

In addition, for any operator $A_{w}$ with $w\in\{0,1\}^{t}$, we let $\prod_{w\in\{0,1\}^{t}}A_{w}\triangleq A_{1^{t}}A_{1^{t-1}0}\cdots A_{0^{t}}$, $\prod^{\prime}_{w\in\{0,1\}^{t}}A_{w}\triangleq A_{0^{t}}A_{0^{t-1}1}$ $\cdots A_{1^{t}}$.

In the following, we prove the correctness of Algorithm 1. The state after the third step of Algorithm 1 is

Then Algorithm 1 queries each of the oracles to get the following state.

After sorting by using $U_{Sort}$, we have the following state.

After that, we query each oracle again and obtain the following state.

After using Hadamard transform on the first register, in the light of Theorem 1, we get the following state.

Note that if there exists $s^{\prime}_{l}\in S_{l}$ such that $s^{\prime}_{l}\cdot z=1$, then we have

If $z\in S_{l}^{\perp}$, then $\sum_{s_{l}\in S_{l}}(-1)^{s_{l}\cdot z}=2^{k_{l}}$, so we have

Thus, in line 9 of Algorithm 1, after measuring the first register of the state $\ket{\psi_{5}}$, we can obtain an element $z\in{S_{l}}^{\perp}$.

In line 9 of Algorithm 1, since the result we measure may not be linearly independent of the results we measured earlier, there is no guarantee that $\langle Y\rangle^{\perp}=S_{l}$ can be obtained for $Y$ obtained after running Algorithm 1 iteratively many times.

After running Algorithm 1 iteratively many times, we denote $S^{\prime}_{l}=\langle Y\rangle^{\perp}$. Denote $k^{\prime}_{l}=rank\left(S^{\prime}_{l}\right)$, $k_{l}\leq k^{\prime}_{l}\leq n-t$, and the basis of $S^{\prime}_{l}$ as $\left\{e^{\prime}_{i}|e^{\prime}_{i}\in\{0,1\}^{n-t},1\leq i\leq k^{\prime}_{l}\right\}$. Note that $S^{\prime}_{l}$ may not be equal to $S_{l}$.

Denote $E_{l}=S_{l}\bigcap\{e^{\prime}_{i}|e^{\prime}_{i}\in\{0,1\}^{n-t},1\leq i\leq k^{\prime}_{l}\}$, $\widehat{k_{l}}=|E_{l}|$, $0\leq\widehat{k_{l}}\leq\min\left\{k^{\prime}_{l},2^{k_{l}}\right\}$.

Denote $S^{\prime}=\bigcup_{j=0}^{2^{\widehat{k_{l}}}-1}\left\{\left(\sum_{i=1}^{\widehat{k_{l}}}\beta_{i}e^{\prime}_{n_{i}}\right)v\Big{|}v\in V_{j}\right\}$. It will be proved below that $S$ may not be equal to $S^{\prime}$, and hence Algorithm 2 is not exact.

In fact, if $S^{\prime}_{l}\neq S_{l}$, then there may exist $e_{i}$ in the basis of $S_{l}$ and $e_{i}\notin\bigcup_{j=0}^{2^{\widehat{k_{l}}}-1}\left\{\sum_{i=1}^{\widehat{k_{l}}}\beta_{i}e^{\prime}_{n_{i}}\Big{|}e^{\prime}_{n_{i}}\in E_{l}\right\}$, where $E_{l}=S_{l}\bigcap\left\{e^{\prime}_{i}|e^{\prime}_{i}\in\{0,1\}^{n-t},1\leq i\leq k^{\prime}_{l}\right\}$, $\widehat{k_{l}}=|E_{l}|$, $\beta_{i}\in\{0,1\}$ and $j=\sum_{i=1}^{\widehat{k_{l}}}2^{i-1}\beta_{i}$. Let $s=e_{i}v_{i}\in S$, where $e_{i}\in S_{l}\setminus\bigcup_{j=0}^{2^{\widehat{k_{l}}}-1}\left\{\sum_{i=1}^{\widehat{k_{l}}}\beta_{i}e^{\prime}_{n_{i}}\Big{|}e^{\prime}_{n_{i}}\in E_{l}\right\}$ and $v_{i}\in S_{r}$. Thus, $s=e_{i}v_{i}\notin S^{\prime}$, which indicates that there is a case where $S$ is not equal to $S^{\prime}$, i.e., Algorithm 2 is not exact.

By means of the work of Cai and Qiu cai_optimal_2018 , we also require a similar method, i.e., the use of quantum amplitude amplification technique to make the distributed quantum algorithm for solving the generalized Simon’s problem exactly.

To make Algorithm 1 exact, we add a post-processing subroutine after line 8 of Algorithm 1 to ensure $(Y\setminus\{0^{n-t}\})\cup\{z\}$ is always linearly independent when we get the measured result $z$ of the first register.

Denote by

and let

denote the state after line 8 of Algorithm 1.

Let $\mathcal{A}:\{0,1\}^{n-t+2^{t+1}m}\rightarrow\{0,1\}^{n-t+2^{t+1}m}$ denote the combined unitary operators from line 4 to line 8 in Algorithm 1, i.e., $\mathcal{A}$ is defined as

Define $\mathcal{R}_{0}(\phi):\{0,1\}^{n-t+2^{t+1}m}\rightarrow\{0,1\}^{n-t+2^{t+1}m}$ as

Define $\mathcal{R}_{\mathcal{A}}(\varphi,Y):\{0,1\}^{n-t}\rightarrow\{0,1\}^{n-t}$ as

Then by $\mathcal{R}_{0}(\phi)$ and $\mathcal{R}_{\mathcal{A}}(\varphi,Y)$, we define the quantum amplitude amplification operator $\mathcal{Q}:\{0,1\}^{n-t+2^{t+1}m}\rightarrow\{0,1\}^{n-t+2^{t+1}m}$ as

Denote by

We have

In order to make Algorithm 1 exact, the crucial step is to eliminate all states in $\langle Y\rangle$ from the first register. In quantum amplitude amplification process, one can achieve this by choosing appropriate $\phi,\varphi\in\mathbb{R}$ such that after applying $\mathcal{Q}$ on $\Ket{{S_{l}}^{\perp},0^{2^{t}m},S(T)}$, the amplitudes of all states in $\langle Y\rangle$ of the first register become zero.

In the following, we present a proposition related to the operator $\mathcal{Q}$ acting on the state $\Ket{{S_{l}}^{\perp},0^{2^{t}m},S(T)}$, which is proved in Appendix A.