Unconditional and exponentially large violation of classicality

TL;DR

This work introduces the complement sampling game as a practical, assumption-light test of quantum non-classicality. By employing BV subset states, the authors prove an unconditional, exponential separation between quantum and classical strategies, with $V(\mathcal{F}_{\mathrm{BV}},Q)=1$ and $V(\mathcal{F}_{\mathrm{BV}},C)=\frac{1}{2^n-1}$, yielding a ratio of $2^n-1$. They implement the game on Quantinuum H2 trapped-ion devices, using quantum teleportation and ancilla-assisted circuits to simulate input channels and the swapper, and observe scores that exceed the classical bound up to $n=37$ (and experimentally explore PRP-based variants). The results substantiate the quantum nature of the hardware under a scalable, efficiently verifiable framework while highlighting remaining loopholes and paths toward fully device-independent tests and fault-tolerant scaling. Overall, the paper presents a concrete, scalable avenue for demonstrating quantum advantage that is robust to some hardware imperfections and does not rely on conjectural complexity-theoretic assumptions.

Abstract

Testing the predictions of quantum mechanics has been one of the main experimental endeavors for decades. Recent advancements in technology led to a number of demonstrations which test non-classicality via specific computational tasks. Limitations of these experiments include dependence on complexity theory assumptions, susceptibility to hardware noise and inefficient verification, raising questions about their scalability. We propose to test non-classicality using a game based on complement sampling, an efficiently verifiable problem that achieves the largest possible separation between quantum and classical computation when both input and output represent samples from probability distributions. When restricting the input to instances inspired by the Bernstein-Vazirani problem, our game admits an exponentially large violation of classicality without relying on computational hardness assumptions. We execute the game on Quantinuum System Model H2 trapped-ion quantum computers, with experiments consisting of thousands of different circuits on up to 55 qubits. The observed scores can be explained by a systematic adoption of a quantum strategy, further corroborating the quantum nature of the hardware in an efficient and scalable way.

Figures (7)

• Figure 1: Complement sampling game for unconditional violation of classicality. (A) At each round, the referee samples a new set $S$ of cardinality $2^{n-1}$, prepares the subset state $\ket{S}$, and sends it to the player. The player chooses a classical or quantum strategy to obtain a sample $y$, supposedly from the complement $\bar{S}$, and sends it to the referee. The referee assigns a score $+1$ if the returned sample is from the complement and $-1$ otherwise. After a number of rounds the referee must determine from the recorded scores if the player used a classical or quantum strategy. (B) Quantum circuit used by the referee to prepare the $n$-qubit Bernstein-Vazirani (BV) subset state $\ket{S}$. Here, $S$ depends on random bit string $u$ and ancilla measurement outcome $b$. The referee verifies $y$ using their knowledge of $u$ and $b$. (C) Teleportation circuit simulating a quantum communication channel between referee and player. This uses $n$ ancillas. (D) The optimal classical strategy is to measure the incoming state in the computational basis and return a different bit string. The swapper circuit implements the unitary $2\dyad{+^n} - I$ and is used by the player adopting the optimal quantum strategy. Additional ancillas can be used to compile this circuit to lower-level gates.
• Figure 1: Numerical analysis of pseudorandom permutations constructions. (a) Total variation distance $\epsilon$ to the uniform distribution versus number of bits $n$. (b) Number of two-qubit $R_{ZZ}$ gates $n_\mathrm{gates}$ required for a reversible circuit implementation optimized for Quantinuum H2-1. Each point is averaged over $100$ random instances. Data corresponding to NN constructions lie directly below the corresponding RT constructions.
• Figure 2: Experimental results. (A) Empirical score for the complement sampling game and state preparation by the referee executed on the H2-2 quantum computer. Outside the shaded region we reject the null hypothesis (that the player used any strategy with expected score at most the optimal classical score) with significance $\alpha=0.01$. (B) Violation of classicality quantified by the ratio of the expected scores for quantum vs classical strategies. Error bars correspond to 99% confidence intervals. The $n=37$ data use $r=500$ rounds, all other data use $r=100$ rounds.
• Figure 2: Experimental results for PRP subset states. Running average of the probability of success plotted against the number of rounds. The optimal classical strategy achieves the probability indicated by the gray line.
• Figure 3: (a) Three-qubit Toffoli gate implemented up to a relative phase and with optimal CX count. (b) Four-qubit Toffoli gate implemented up to a relative phase. (c) Four-qubit Toffoli and (d) five-qubit Toffoli implemented using a single ancilla qubit. RTOF3 is self-adjoint, but we keep the $\dagger$ symbol as a guidance for the recursive construction of high-order Toffoli gates explained in the text. All circuits are taken from Ref. Maslov_2016.

Theorems & Definitions (9)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3: restatement of Thm. 7 in Ref. benedetti2025complement
• Definition 1: restatement of Def. 1 in Ref. gay2025pseudorandomness
• Theorem 4: restatement of Thm. 3 in Ref. gay2025pseudorandomness
• Theorem 5: restatement of Thm. 3 in Ref. diakonikolas2017optimal
• Theorem 6: restatement of Prop. 4 in Ref. Maslov_2016