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Average-case complexity versus approximate simulation of commuting quantum computations

Michael J. Bremner, Ashley Montanaro, Dan J. Shepherd

TL;DR

It is shown that, if either of two plausible average-case hardness conjectures holds, then IQP computations are hard to simulate classically up to constant additive error.

Abstract

We use the class of commuting quantum computations known as IQP (Instantaneous Quantum Polynomial time) to strengthen the conjecture that quantum computers are hard to simulate classically. We show that, if either of two plausible average-case hardness conjectures holds, then IQP computations are hard to simulate classically up to constant additive error. One conjecture relates to the hardness of estimating the complex-temperature partition function for random instances of the Ising model; the other concerns approximating the number of zeroes of random low-degree polynomials. We observe that both conjectures can be shown to be valid in the setting of worst-case complexity. We arrive at these conjectures by deriving spin-based generalisations of the Boson Sampling problem that avoid the so-called permanent anticoncentration conjecture.

Average-case complexity versus approximate simulation of commuting quantum computations

TL;DR

It is shown that, if either of two plausible average-case hardness conjectures holds, then IQP computations are hard to simulate classically up to constant additive error.

Abstract

We use the class of commuting quantum computations known as IQP (Instantaneous Quantum Polynomial time) to strengthen the conjecture that quantum computers are hard to simulate classically. We show that, if either of two plausible average-case hardness conjectures holds, then IQP computations are hard to simulate classically up to constant additive error. One conjecture relates to the hardness of estimating the complex-temperature partition function for random instances of the Ising model; the other concerns approximating the number of zeroes of random low-degree polynomials. We observe that both conjectures can be shown to be valid in the setting of worst-case complexity. We arrive at these conjectures by deriving spin-based generalisations of the Boson Sampling problem that avoid the so-called permanent anticoncentration conjecture.

Paper Structure

This paper contains 6 sections, 9 theorems, 33 equations, 1 figure.

Table of Contents

  1. The Ising model and IQP circuits
  2. Gaps of degree-3 polynomials
  3. The Hadamard gadget
  4. Hardness of multiplicative approximations
  5. From additive to multiplicative error
  6. Proofs of anticoncentration bounds

Key Result

Theorem 1

Assume either Conjecture con:Ising or con:deg3 below is true. If it is possible to classically sample from the output probability distribution of any IQP circuit $\mathcal{C}$ in polynomial time, up to an error of 1/192 in $\ell_1$ norm, then there is a $\text{BPP}^{\text{NP}}$ algorithm to solve an

Figures (1)

  • Figure 1: An example of a randomly chosen circuit $\mathcal{C}_I$ corresponding to a 4-qubit Ising model instance such that $\langle 0|^{\otimes n} \mathcal{C}_I|0\rangle^{\otimes n} = Z_R/2^n$ (up to trivial phase factors). Assuming Conjecture \ref{['con:Ising']} is true, if there exists a classically efficient algorithm for sampling from the output of any such ($n$-qubit) circuit to within a constant additive error, then the Polynomial Hierarchy collapses.

Theorems & Definitions (17)

  • Theorem 1
  • Conjecture 2
  • Conjecture 3
  • Lemma 4
  • proof
  • Theorem 6
  • proof
  • Theorem 7
  • Proposition 8
  • proof
  • ...and 7 more