Complexity-Theoretic Foundations of Quantum Supremacy Experiments

TL;DR

The paper builds a rigorous complexity-theoretic framework for quantum supremacy experiments, focusing on sampling tasks from random quantum circuits and the verification of outputs. It introduces the Heavy Output Generation task and the QUATH hardness assumption, then presents a Savitch-style polynomial-space classical simulator and grid-based speedups to probe classical limits. It shows that strong supremacy theorems must be non-relativizing and provides maximal quantum-classical gaps for Fourier-based problems, along with oracle-relative results that connect quantum advantage to computational assumptions such as one-way functions. The work clarifies the demands high-depth, large-gate-count circuits impose for robust supremacy signals and outlines a smooth spectrum of hardness results tied to different computational models, including P/poly oracles, with implications for near-term quantum experiments.

Abstract

In the near future, there will likely be special-purpose quantum computers with 40-50 high-quality qubits. This paper lays general theoretical foundations for how to use such devices to demonstrate "quantum supremacy": that is, a clear quantum speedup for some task, motivated by the goal of overturning the Extended Church-Turing Thesis as confidently as possible. First, we study the hardness of sampling the output distribution of a random quantum circuit, along the lines of a recent proposal by the the Quantum AI group at Google. We show that there's a natural hardness assumption, which has nothing to do with sampling, yet implies that no efficient classical algorithm can pass a statistical test that the quantum sampling procedure's outputs do pass. Compared to previous work, the central advantage is that we can now talk directly about the observed outputs, rather than about the distribution being sampled. Second, in an attempt to refute our hardness assumption, we give a new algorithm, for simulating a general quantum circuit with n qubits and m gates in polynomial space and m^O(n) time. We then discuss why this and other known algorithms fail to refute our assumption. Third, resolving an open problem of Aaronson and Arkhipov, we show that any strong quantum supremacy theorem--of the form "if approximate quantum sampling is classically easy, then PH collapses"--must be non-relativizing. Fourth, refuting a conjecture by Aaronson and Ambainis, we show that the Fourier Sampling problem achieves a constant versus linear separation between quantum and randomized query complexities. Fifth, we study quantum supremacy relative to oracles in P/poly. Previous work implies that, if OWFs exist, then quantum supremacy is possible relative to such oracles. We show that some assumption is needed: if SampBPP=SampBQP and NP is in BPP, then quantum supremacy is impossible relative to such oracles.

Figures (5)

• Figure 1: $\mathcal{C}_1 \to \mathcal{C}_2$ indicates $\mathcal{C}_1$ is contained in $\mathcal{C}_2$ respect to every oracle in $\textsf{P/poly}$, and $\mathcal{C}_1 \dashrightarrow \mathcal{C}_2$ denotes that there is an oracle $\mathcal{O} \in \textsf{P/poly}$ such that $\mathcal{C}_1^{\mathcal{O}} \not\subset \mathcal{C}_2^{\mathcal{O}}$. Red indicates this statement is based on the existence of classical one-way functions, Blue indicates the statement is based on the existence of quantum one-way functions, and Black indicates the statement holds unconditionally.
• Figure 2: A histogram of (normalized) $\mathsf{probList}(C|0\rangle)$, where $C \leftarrow \mu_{\mathsf{grid}}^{16,256}$. The x-axis represents the probability, and the y-axis represents the estimated density, and the red line indicates the PDF of the exponential distribution with $\lambda = 1$.
• Figure 3: A histogram of the $\mathsf{adv}(C)$'s of the $10^5$ i.i.d. samples from $\mu_{\mathsf{grid}}^{9,81}$. The x-axis represents the value of $\mathsf{adv}(C)$, and the y-axis represents the estimated density, and the red line indicates the PDF of the normal distribution $\mathcal{N}(0.846884, 0.00813911^2)$.
• Figure 4: A histogram of the $\mathsf{adv}(C)$'s of the $10^5$ i.i.d. samples from $\mu_{\mathsf{grid}}^{16,256}$. The x-axis represents the value of $\mathsf{adv}(C)$, the y-axis represents the estimated density, and the red line indicates the PDF of the normal distribution $\mathcal{N}(0.846579, 0.000712571^2)$.
• Figure 5: The empirical decay of the variance of $\mathsf{adv}(C)$. Here a point $(x,y)$ means that the standard variance of the corresponding $\mathsf{adv}(C)$'s for the 1000 i.i.d. samples from $\mu_{\mathsf{general}}^{x,x^2}$ is $y$. Also, the red line represents the function $y = 0.1/x$.

Theorems & Definitions (83)

• Definition 2.1
• Definition 2.2
• Definition 2.3: Sampling Problems, $\textsf{SampBPP}$, and $\textsf{SampBQP}$
• Theorem 2.4: Helstrom's decoder for two pure states
• Corollary 2.5
• proof
• Lemma 2.6
• Corollary 2.7
• proof
• Lemma 3.1