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Bell Sampling

System-level

Two-copy benchmark that measures pairs of identical circuit outputs in the Bell basis, yielding fidelity estimates and circuit diagnostics that stay efficient beyond the classically simulable regime.

Measures
State fidelity of an n-qubit circuit output, estimated from transversal Bell-basis measurements on two copies prepared in parallel; the same samples give a depth test and a T-count lower bound
Introduced by
Hangleiter & Gullans (QuICS, NIST / University of Maryland) (2023)
Also known as
Bell sampling from quantum circuits

Bell sampling is a benchmarking and verification protocol introduced by Dominik Hangleiter and Michael Gullans in 2023: prepare the output state of an n-qubit circuit twice in parallel and measure the two copies against each other in the Bell basis. The samples estimate the output-state fidelity directly, and verification stays efficient beyond the classically simulable regime where cross-entropy benchmarking needs classical simulation, so the protocol is positioned as XEB’s successor. The paper was published as Phys. Rev. Lett. 133, 020601 (2024).

How it works

Two identical copies of the circuit’s output state are prepared side by side, occupying 2n qubits. A single transversal layer then measures qubit i of one copy jointly with qubit i of the other in the two-qubit Bell basis. Hangleiter and Gullans call the resulting dataset a “circuit shadow”: from the same samples one can estimate the state fidelity, test the circuit’s depth, lower-bound its T-count, and diagnose certain preparation errors. Because Bell samples are argued to be classically intractable to produce, the protocol doubles as an efficiently validated quantum-advantage scheme.

Strengths and limitations

Verification is sample-efficient with no exponential classical cost, and one dataset yields several diagnostics at once. The price is hardware: an n-qubit benchmark occupies 2n qubits plus a transversal two-qubit measurement layer, doubling the requirements of single-copy schemes like XEB. It produces fidelity estimates and property tests rather than a single branded score, and it is not yet a routine vendor-reported number the way XEB or Quantum Volume are. The name predates the benchmark: Montanaro (2017) used “Bell sampling” for learning stabilizer states, a result the 2023 protocol builds on. The naive extension to qudits also fails, requiring a modified unitary. No public reference implementation was found.

Notable results

Bell-sampling measurements of quantum shadow enumerators ran on a trapped-ion quantum computer (Phys. Rev. Research 8, 023318, 2026), qudit generalizations appeared in 2025, and Hangleiter’s 2026 review Has quantum advantage been achieved? presents Bell-basis measurement of two circuit copies as a route to closing the verification loophole in quantum-advantage claims.

Key papers

  • Cross-Entropy Benchmarking : Estimates circuit fidelity by checking how often a device samples the high-probability bitstrings of random quantum circuits, the metric behind Google's quantum supremacy claim.