Bell sampling from quantum circuits

TL;DR

Bell sampling introduces a universal quantum-computation model based on measuring two copies of a circuit's output in the transversal Bell basis, producing samples that are classically hard to simulate yet encode rich diagnostic information about the circuit. The authors prove universality, provide complexity-theoretic evidence for classical hardness, and develop practical protocols to extract circuit properties, including fidelity estimation, depth testing, and learning of low-$T$ Clifford+$T$ circuits, treating Bell samples as a classical shadow of the circuit. They also demonstrate error-detection/correction advantages, discuss robustness to noise, and adapt existing noisy-simulation frameworks to Bell sampling, highlighting both the potential for near-term quantum advantage demonstrations and the challenges of efficient classical simulation under realistic noise. The work positions Bell sampling as a realistic, benchmarkable pathway toward fault-tolerant quantum-computation verification with direct classical validation from Bell-sample data, while outlining open questions about noise robustness and scalable simulation.

Abstract

A central challenge in the verification of quantum computers is benchmarking their performance as a whole and demonstrating their computational capabilities. In this work, we find a universal model of quantum computation, Bell sampling, that can be used for both of those tasks and thus provides an ideal stepping stone towards fault-tolerance. In Bell sampling, we measure two copies of a state prepared by a quantum circuit in the transversal Bell basis. We show that the Bell samples are classically intractable to produce and at the same time constitute what we call a circuit shadow: from the Bell samples we can efficiently extract information about the quantum circuit preparing the state, as well as diagnose circuit errors. In addition to known properties that can be efficiently extracted from Bell samples, we give several new and efficient protocols: an estimator of state fidelity, a test for the depth of the circuit and an algorithm to estimate a lower bound to the number of T gates in the circuit. With some additional measurements, our algorithm learns a full description of states prepared by circuits with low T-count.

Figures (5)

• Figure 1: The Bell sampling protocol. In the Bell sampling protocol we prepare the quantum state $C \ket {0^n}\otimes C \ket {0^n}$ using a quantum circuit $C$, and measure all qubits transversally in the Bell basis across the bipartition of the system.
• Figure 2: Fidelity estimation based on noisy Bell sampling. We simulate noisy Bell sampling and XEB including noisy measurements using $10^6$ samples and compute the fidelity (lines), and the purity (hexagons), and XEB (crosses) based estimators of fidelity for (a) typical Clifford circuits with two-qubit random gates in an all-to-all connected architecture with XEB $1$ on $n = 20$ qubits and Pauli $(X,Y,Z)$ error probabilies $p = 0.005 \cdot (1, 1/3,1/10)$, and (b) crystalline Floquet Clifford circuits that are scrambling sommers_crystalline_2023 on $18$ qubits in 1D with (depolarized) two-qubit gate fidelity $0.98$. Missing XEB points are due to ideal XEB values $0$. Error bars represent one standard deviation.
• Figure 3: Depth-dependent Page curves. (a) The maximal subsystem entanglement entropy depends on the circuit architecture and depth (shades of blue) until the half-cut entanglement reaches its maximal value given by $n/2$. We measure the subsystem entropy at half-cuts to obtain the maximal sensitivity to different circuit depths. (b) We detect errors in the Bell samples by detecting strings that lead to a non-zero estimate of the purity of $\rho$.
• Figure S4: Performance of the Bell sampling fidelity estimator. To assess the purity estimator, we simulate random and non-random Clifford circuits and perform direct fidelity estimation, Bell sampling, and standard-basis sampling to compute the fidelity (green lines), root purity (blue hexagons), and normalized linear XEB (pink crosses), respectively, each obtained from $10^6$ samples per circuit. For the purpose of this plot, the Bell measurement is assumed to be noise-free. Error bars represent one standard deviation. (a) Average fidelity over 100 random two-qubit Clifford circuits for $n = 6,12,18,24,30,36$ qubits (in decreasing opacity) in an all-to-all connected architecture with $12$ gate layers with local depolarizing noise. (b) Single-instance of a random circuit for $n = 6,12,18,24,30,36$ qubits (decreasing opacity) with $12$ gate layers in an all-to-all connected architecture with local Pauli noise with $(X,Y,Z)$ error probabilites $(p, p/3,p/10)$. (c) Non-scrambling 1D, depth-$2$ crystalline circuit with iSWAP entangling gatesfrom Ref. sommers_crystalline_2023 on 18 qubits with no single-qubit gates. (d) Scrambling 1D, depth-$2$ crystalline circuit with iSWAP entangling gatesfrom Ref. sommers_crystalline_2023 on 18 qubits with single-qubit $\sqrt{X}$ gates following every two-qubit gate.
• Figure S5: Fidelity estimation based on noisy Bell sampling. We simulate noisy Bell sampling and XEB including noisy measurements using $10^7$ samples. (a) Single-instance fidelity (lines), purity estimator (hexagons), and XEB (crosses) of typical random Clifford circuits for $n = 6,12,18,24,30$ (in decreasing opacity) with depth $12$ and Pauli $(X,Y,Z)$ error probabilies $p (1,1/3,1/10)$. (b) Fidelity of non-random scrambling crystalline circuits sommers_crystalline_2023 of depth $2$ acting on $18$ qubits in 1D.

Theorems & Definitions (10)

• Lemma 1: -completeness
• proof
• Lemma 2
• proof : Proof
• Lemma 3: Translating fidelity and purity
• proof
• Lemma 4
• proof
• Lemma 5: Weighted subspace generation
• proof