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Classical simulability of quantum circuits followed by sparse classical post-processing

Yasuhiro Takahashi, Masayuki Miyamoto, Noboru Kunihiro

TL;DR

It is shown that it is simulable by a polynomial-time probabilistic algorithm with access to commuting quantum circuits on n+1 qubits, which provides a better understanding of the hardness of simulating constant-depth quantum circuits followed by SCP.

Abstract

We study the classical simulability of a polynomial-size quantum circuit $C_n$ on $n$ qubits followed by sparse classical post-processing (SCP) on $m$ bits, where $m \leq n \leq {\rm poly}(m)$. The SCP is described by a non-zero Boolean function $f_m$ that is classically computable in polynomial time and is sparse, i.e., has a peaked Fourier spectrum. First, we provide a necessary and sufficient condition on $C_n$ such that, for any SCP $f_m$, $C_n$ followed by $f_m$ is classically simulable. This characterization extends the result of Van den Nest and implies that various quantum circuits followed by SCP are classically simulable. Examples include IQP circuits, Clifford Magic circuits, and the quantum part of Simon's algorithm, even though these circuits alone are hard to simulate classically. Then, we consider the case where $C_n$ has constant depth $d$. While it is unlikely that, for any SCP $f_m$, $C_n$ followed by $f_m$ is classically simulable, we show that it is simulable by a polynomial-time probabilistic algorithm with access to commuting quantum circuits on $n+1$ qubits. Each such circuit consists of at most deg($f_m$) commuting gates and each commuting gate acts on at most $2^d+1$ qubits, where deg($f_m$) is the Fourier degree of $f_m$. This provides a better understanding of the hardness of simulating constant-depth quantum circuits followed by SCP.

Classical simulability of quantum circuits followed by sparse classical post-processing

TL;DR

It is shown that it is simulable by a polynomial-time probabilistic algorithm with access to commuting quantum circuits on n+1 qubits, which provides a better understanding of the hardness of simulating constant-depth quantum circuits followed by SCP.

Abstract

We study the classical simulability of a polynomial-size quantum circuit on qubits followed by sparse classical post-processing (SCP) on bits, where . The SCP is described by a non-zero Boolean function that is classically computable in polynomial time and is sparse, i.e., has a peaked Fourier spectrum. First, we provide a necessary and sufficient condition on such that, for any SCP , followed by is classically simulable. This characterization extends the result of Van den Nest and implies that various quantum circuits followed by SCP are classically simulable. Examples include IQP circuits, Clifford Magic circuits, and the quantum part of Simon's algorithm, even though these circuits alone are hard to simulate classically. Then, we consider the case where has constant depth . While it is unlikely that, for any SCP , followed by is classically simulable, we show that it is simulable by a polynomial-time probabilistic algorithm with access to commuting quantum circuits on qubits. Each such circuit consists of at most deg() commuting gates and each commuting gate acts on at most qubits, where deg() is the Fourier degree of . This provides a better understanding of the hardness of simulating constant-depth quantum circuits followed by SCP.
Paper Structure (17 sections, 11 theorems, 82 equations, 3 figures)

This paper contains 17 sections, 11 theorems, 82 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and main results
  3. Overview of techniques
  4. Preliminaries
  5. Quantum circuits
  6. Fourier expansions of Boolean functions
  7. Quantum circuits followed by SCP
  8. General quantum circuits followed by SCP
  9. Hardness of constant-depth quantum circuits followed by SCP
  10. Proof of (1) => (2) in Theorem \ref{['main-theorem']}
  11. Proof of (2) => (1) in Theorem \ref{['main-theorem']}
  12. Applications of Theorem \ref{['main-theorem']}
  13. CT states and ECS operations
  14. IQP circuits and the Quantum part of Simon's algorithm
  15. Clifford Magic circuits
  16. ...and 2 more sections

Key Result

Theorem 1

Let $C_n$ be an arbitrary polynomial-size quantum circuit on $n$ qubits initialized to $|0^n\rangle$, and $m$ be an arbitrary integer such that $m \leq n \leq {\rm poly}(m)$. The following items are equivalent:

Figures (3)

  • Figure 1: $C_n$ followed by $f_m$. The circuit $C_n$ is applied to the initial state $|0^n\rangle$, the top $m$ qubits are measured in the computational basis to obtain a classical outcome $x= x_1\cdots x_m \in \{0,1\}^m$, and the non-zero Boolean function $f_m$ is applied to $x$, yielding the output $f_m(x) \in \{0,1\}$.
  • Figure 2: (a) Quantum circuit for an efficiently computable basis-preserving operation sandwiched between two Hadamard layers. Examples of such circuits are IQP circuits and the quantum part of Simon's algorithm. (b) Clifford Magic circuit.
  • Figure 3: Quantum circuit for approximating $\langle 0^n |C_n^\dagger Z(1^n) C_n |0^n\rangle$. The right hand side of the equation is a commuting quantum circuit whose output probability distribution is the same as that of the left hand side.

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • proof
  • ...and 7 more