Random Circuit Sampling (RCS)

• Random Circuit Sampling (RCS) is a method that samples outcomes from random quantum circuits to benchmark quantum advantage and certify randomness.
• It utilizes complexity-theoretic reductions and anti-concentration properties, such as the Porter–Thomas law, to underline computational hardness.
• RCS plays a critical role in experimental validations on platforms like Google's Sycamore and USTC’s Zuchongzhi, driving progress in noise analysis and quantum randomness certification.

Random Circuit Sampling (RCS) is the problem of sampling computational-basis bitstrings from the output distribution of a quantum circuit whose gates are drawn randomly from a specified ensemble, typically Haar-random unitaries over a fixed (usually sparse, local) architecture. Designed as a canonical candidate for demonstrating quantum computational advantage, RCS sits at the intersection of complexity theory, quantum chaos, and experimental benchmarking, underpinning benchmark experiments such as Google’s Sycamore and USTC’s Zuchongzhi. The complexity-theoretic, statistical, and physical features of RCS have led to its adoption as both a practical benchmark and a foundational object for certifying quantum randomness, analyzing noise, and studying quantum/classical hardness.

1. Definition and Standard Protocols

In the typical protocol, $n$ qubits are initialized in the state $|0^n\rangle$, and a depth-$d$ quantum circuit $C$ is drawn by concatenating $d$ layers of one- and two-qubit gates, often sampled from the Haar measure or an approximate unitary $2$-design, laid out on a fixed connectivity graph (e.g., 2D grid). The output distribution is given by

$$D_C(s) = |\langle s|C|0^n\rangle|^2,\quad s\in\{0,1\}^n.$$

The RCS task is, given a description of $C$, to generate samples $s$ drawn from $D_C$ (Bassirian et al., 2021, Bouland et al., 2018, Movassagh, 2019, Oh et al., 2022). In the infinite-depth, Haar-random limit, the probability distribution over bitstrings becomes exponentially distributed (Porter–Thomas law), setting statistical signatures for chaotic quantum circuits (Zhu et al., 2021).

In the context of benchmarking and certification, the linear cross-entropy benchmarking fidelity (XEB) is the empirical metric:

$|0^n\rangle$0

where $|0^n\rangle$1 is the ideal probability and the average is over samples from the device (Liu et al., 2021).

2. Complexity-Theoretic Foundation and Hardness

RCS’s central theoretical role is as a quantum advantage witness rooted in the conjectured classical intractability of output sampling. For a broad class of architectures, worst-case output probability computation (e.g., $|0^n\rangle$2) is $|0^n\rangle$3-hard (Bouland et al., 2018, Movassagh, 2019). Worst-to-average-case reductions, leveraging polynomial or rational-function interpolation (geodesic, Cayley, or QR-path interpolations, combined with Berlekamp–Welch decoding), show that computing these probabilities exactly, even on most random instances, remains $|0^n\rangle$4-hard (Bouland et al., 2018, Movassagh, 2018, Movassagh, 2019).

Explicitly, even additive estimation within $|0^n\rangle$5, with $|0^n\rangle$6 the circuit size, is $|0^n\rangle$7-hard for a $|0^n\rangle$8 fraction of Haar-random circuits (Movassagh, 2019). This forms the basis of the extended Church–Turing thesis challenge.

Approximate-average-case hardness for output probabilities to inverse-polynomial errors, essential for direct cryptographic or sampling hardness, remains an open question (Movassagh, 2018). The anti-concentration property—probabilities not being concentrated near zero—is established for sufficiently deep 1D/2D random circuits as a direct consequence of local circuits forming approximate unitary 2-designs (Bouland et al., 2018, Morvan et al., 2023). This property links Stockmeyer-type counting to the average-case hardness reduction for sampling (Bouland et al., 2018).

3. Verification, Certification, and Randomness Generation

Verification of RCS-based devices relies both on statistical properties (Porter–Thomas law, heavy-output generation (HOG), NIST tests, cross-entropy, Marchenko–Pastur law, Wasserstein distances) and protocolized approaches. RCS’s structure supplies the basis for certified randomness protocols. In Aaronson’s RCS-based certified randomness proposal, a classical verifier selects random circuits $|0^n\rangle$9 and tasks a quantum device to produce one sample per circuit. For certification, the “HOG” test checks that at least $d$0 of outputs are “heavy” (probability $d$1); passing implies high output min-entropy contingent on the Long-List Quantum Supremacy Verification (LLQSV) conjecture (Bassirian et al., 2021).

Bassirian et al. (Bassirian et al., 2021) render a black-box Fourier Sampling analogue in the Quantum Random Oracle Model (QROM), supplying unconditional (albeit black-box) min-entropy lower bounds via the “Fourier HOG” (FHOG) test. Their analysis demonstrates that no $d$2-query algorithm can both pass FHOG and output $d$3 min-entropy. This supports the conjectured entropy-soundness of the original RCS HOG protocol.

Verification methods in hardware experiments employ linear-XEB and collision-based metrics (collision anomaly, collision volume, cross-collision volume) (Mari, 2023), with full statistical analysis extending to spatial correlations and higher-moment distributional properties (Oh et al., 2022). Recent theoretical work leverages high-dimensional mixture modeling to extract spatiotemporal error profiles from experimental RCS data, reaching information-theoretic limits on learnability in various regimes of side information (Manole et al., 10 Oct 2025).

4. Noise, Anti-Concentration, and Classical Simulability

The role of noise is critical in regulating both the theoretical hardness and practical quantum advantage of RCS. For circuits with local, unital (e.g., depolarizing) noise per gate, Fourier–Pauli expansions reveal exponential decay of higher-weight paths (Aharonov et al., 2022, Liu et al., 2021). In this regime, classical polynomial-time sampling algorithms efficiently approximate the noisy output distribution up to inverse-polynomial total variation distance, showing that constant per-gate noise precludes scalable experimental violations of the Church–Turing thesis using RCS (Aharonov et al., 2022). The anti-concentration property remains intact up to moderate depths, but falls off as noise dominates.

By contrast, for non-unital noise such as amplitude damping—pervasive in experimental platforms—anti-concentration fails entirely: the output never flattens, regardless of circuit depth (Fefferman et al., 2023). In this regime, the Porter–Thomas tail is absent and neither hardness nor classical simulability follows immediately from previous reductions, necessitating new techniques.

Empirical RCS implementations (Google Sycamore, USTC Zuchongzhi, IBM, and others) have reported quantum advantage by executing large-scale, moderate-depth random circuits, benchmarking performance via XEB and related protocols (Oh et al., 2022, Zhu et al., 2021, Kasirajan et al., 2024). However, successive advances in classical tensor-network simulators, contraction algorithms, and resource-efficient GPU implementations have rapidly narrowed or erased the practical quantum/classical separation for modest circuit sizes and depths, especially when noise limits XEB fidelity (Zhao et al., 2024, Kasirajan et al., 2024).

5. Scaling, Circuit Architecture, and the Boundary of Quantum Advantage

RCS complexity scales steeply with both width ($d$4) and depth ($d$5), with classical simulation costs driven by memory (state-vector, $d$6) and time (tensor contraction, $d$7 for 2D lattices) (Zhu et al., 2021, Zlokapa et al., 2020). The quantum runtime to achieve fixed-precision XEB decays as $d$8, so the decay of fidelity with circuit size and depth—$d$9, $C$0 being the effective error per layer—defines a crossover: for sufficiently large $C$1, classical simulation becomes more efficient than noisy quantum sampling (Zlokapa et al., 2020).

Table: Representative thresholds (Quantum vs. Classical Runtime), summarized from (Zlokapa et al., 2020)

Platform	Qubits n	Depth d	XEB	Estimated Classical Cost	Quantum Cost
Sycamore	53	20	0.2	~16 days (2019)	600 s
Zuchongzhi 2.1	60	24	0.00037	~4.8 × 10⁴ years	4.2 h

Classical advantage re-emerges at large depths due to the exponential penalty from decaying XEB, delimiting a “supremacy window” in the (n, d) plane whose size is set by error rates and circuit structure. Lowering gate and measurement errors (e.g., towards 0.1%) expands this window, potentially overlapping with early surface code logical qubits.

Recent work introduced "holographic" RCS (HRCS), interleaving mid-circuit measurements and resets to extend the effective sampling dimension exponentially in depth, demonstrating substantial increases in effective circuit complexity on hardware with fixed qubit count (Zhang et al., 7 Nov 2025).

6. Statistical Certification, Alternative Metrics, and Current Limitations

Statistical validation of RCS outputs leverages XEB but also extends to multi-metric fingerprints, including the Marchenko–Pastur eigenvalue spectrum of sample matrices, Wasserstein distances, and heat-map analyses of bitstring observables and spatial or temporal patterns (Oh et al., 2022). Single-metric certification (XEB alone) is insufficient: classical spoofing strategies and Monte Carlo samplers drawing from the Haar distribution (but not from any physical circuit) can achieve nearly maximal XEB at negligible computational cost (Raab, 4 Sep 2025). Full statistical or structural certification remains an open challenge for robust advantage claims.

Collision tests and cross-collision benchmarking provide additional device-insensitive diagnostics, though with exponentially scaling quantum sample cost (Mari, 2023). The ability to extract error profiles, detect correlated/contextual errors, and not merely scalar fidelity, from large-scale RCS data is an area of active research (Manole et al., 10 Oct 2025).

7. Outlook and Open Problems

While the complexity-theoretic evidence for average-case hardness of exact RCS is strong (Bouland et al., 2018, Movassagh, 2019, Movassagh, 2018), robust average-case hardness for approximate sampling remains unproven. The boundary of practical quantum advantage, especially as classical simulators scale and hardware errors remain non-negligible, is responsive to technological progress on both fronts (Zhao et al., 2024, Kasirajan et al., 2024, Zhu et al., 2021). For non-unital noise models and highly realistic error sources, the structure and hardness of RCS output distributions are only partially understood (Fefferman et al., 2023).

Key open questions include:

• Proving approximate-average-case hardness for sampling from random circuit output distributions under realistic noise and architectural constraints.
• Certifying generic min-entropy or randomness generation from RCS protocols (including unconditional security without exponential-time verification) (Bassirian et al., 2021).
• Scaling quantum advantage benchmarks to the regime of error-corrected qubits or leveraging novel circuit constructions (e.g., HRCS) to tighten separation from classical feasibility (Zhang et al., 7 Nov 2025).
• Expanding practical, device-free statistical certification protocols capable of robustly detecting quantum vs. classical or unitarily-generated vs. “spoof” samples under all operationally realistic scenarios.

RCS continues to serve as both a practical and foundational tool for quantum benchmarking, complexity-theoretic validation, and randomness certification, while driving innovation in quantum experiment, classical simulation, and the theory of quantum computational supremacy.