Provable and Verifiable Quantum Advantage in Sample Complexity

TL;DR

This work studies a fundamental quantum-classical separation in the realm of sample complexity by focusing on complement sampling: given a quantum sample from a subset $S$ of $\omega=\{0,1\}^n$ with $|S|=K$, output a sample from the complement $\bar{S}$. The authors show a quantum procedure that converts $\ket{S}$ into $\ket{\bar{S}}$ using a single quantum sample, yielding a sample from $\bar{S}$ with probability $\frac{\min\{K,N-K\}}{\max\{K,N-K\}}$, and, in the special case $K=N/2$, achieves success probability 1—while any classical approach with bounded error requires $\Omega(N)$ samples. They connect the swapper problem to distinguishability of conjugate phase states, proving a no-go result for a perfect swapper when $K\neq N/2$, and they present a Las Vegas (zero-error) stochastic swapper that is optimal under a single ancilla, with resource- and copy-usage analyses. The classical side establishes tight lower and upper bounds in an index-query model, transfers worst-case hardness to average-case via random permutations, and shows strong cryptographic hardness under one-way functions through strong pseudorandom permutations; a Kolmogorov-based argument yields exponential circuit lower bounds for uniform samplers. Collectively, the results justify a provable, verifiable quantum advantage in a sample-to-sample setting and outline a practical path toward NISQ demonstrations of quantum superiority in sampling tasks.

Abstract

Consider a fixed universe of $N=2^n$ elements and the uniform distribution over elements of some subset of size $K$. Given samples from this distribution, the task of complement sampling is to provide a sample from the complementary subset. We give a simple quantum algorithm that uses only a single quantum sample -- a single copy of the uniform superposition over elements of the subset. When $K=N/2$, we show that the quantum algorithm succeeds with probability $1$, whereas any classical algorithm that succeeds with bounded probability of error requires a number of samples of the order of $N$. This shows that in a sample-to-sample setting, quantum computation can achieve the largest possible separation over classical computation. We show that the same bound can be lifted to prove average-case hardness, paving the way for demonstrations on noisy intermediate-scale quantum (NISQ) computers. It follows that under the assumption of the existence of one-way functions, complement sampling gives provable, verifiable and NISQable quantum advantage in a sample complexity setting.

Figures (6)

• Figure 1: Distinguisher and swappers for complement sampling. (a) Our distinguisher can be thought of as the Deutsch-Jozsa algorithm followed by a copy of the measurement outcome into an ancilla qubit. (b) The complement swapper can be thought of as the Grover diffusion operator $U = 2 \dyad{+^n} - \mathbb{I}$ up to a global phase. (c) The zero-error swapper uses the controlled-$U$, the gate $W(q) = e^{i \arccos(\sqrt{q}) Y}$, and an ancilla qubit that upon measurement outcome 0 indicates correctness. The parameters are set to $b = 0$, $q = \frac{1}{2(1 - K/N)}$ if $K < \frac{N}{2}$, and to $b = 1$, $q = \frac{N}{2K}$ if $K \geq \frac{N}{2}$.
• Figure 2: Comparison of complement sampling algorithms given a single sample as input. The vertical axis is the probability of obtaining a sample from the complementary set. The horizontal axis is the deviation from the ideal subset size ratio $K/N = 1/2 + \beta$. When $\beta=0$ the quantum algorithms based on swappers are exact.
• Figure 3: A single round of the complement sampling game with $j$ input samples. The referee picks subset $S$ from a family of subsets of fixed cardinality $\mathcal{S}$. The quantum player gets access to $j$ copies of the state $\ket{S}$, and the classical player gets access to $j$ elements sampled according to the uniform distribution over $S$. Giving the classical player access to measurements of $\ket{S}$ in the computational basis would yield the same input model. Giving the quantum player access to classical samples does not yield any advantage over the classical player. Each player sends back a candidate element $\hat{y}$ and the referee checks whether it belongs to the complementary subset $\bar{S}$.
• Figure 4: Quantum circuits for Aaronson, Atia and Susskind’s construction aaronson2020hardness. (a) Circuit distinguishing $\ket{\phi^+}$ from $\ket{\phi^-}$ using a unitary $U$ that swaps $\ket{a}$ and $\ket{b}$. (b) Circuit swapping $\ket{a}$ and $\ket{b}$ using a unitary $A$ that distinguishes $\ket{\phi^+}$ and $\ket{\phi^-}$.
• Figure 5: Distinguisher and swapper for complement sampling. (a) Our distinguisher circuit can be thought of as the Deutsch-Jozsa algorithm followed by a copy of the measurement outcome into an ancilla qubit. (b) The swapper circuit is obtained by plugging our distinguisher circuit into \ref{['fig:aas_construction']} (b) and simplifying gates. The simplification leads to the ancilla qubit being idle, so we do not include it in this circuit diagram. Our swapper circuit corresponds to the Grover diffusion operator up to a global phase of $-1$.

Theorems & Definitions (40)

• Theorem 1: Informal, from Theorems 5 and 6 in SM
• Theorem 2: Informal, from Theorem 8 in SM
• Theorem 3: Informal, from Theorem 7 in SM
• Definition 1: Relative complexity
• Definition 2: Circuit complexity
• Definition 3: Swap complexity
• Definition 4: Distinguishability complexity
• Lemma 1: Adapted from aaronson2020hardness, Theorem 2
• Proposition 1
• proof