Clifford Volume and Free Fermion Volume: Complementary Scalable Benchmarks for Quantum Computers

TL;DR

This work introduces two complementary volumetric benchmarks, Clifford Volume (CLV) and Free Fermion Volume (FFV), to quantify quantum hardware capacity in a scalable, classically verifiable, and platform-agnostic way. CLV tests random Clifford unitaries via stabilizer/destabilizer Pauli measurements, while FFV tests free-fermion evolutions mapped through Jordan–Wigner, enabling orthogonality checks with Majorana operators; together they span a universal gate set and reflect practical primitives. The authors provide rigorous threshold-based verification criteria, baseline realizations, and numerical studies under realistic noise, plus an experimental demonstration of CLV achieving 34 on Quantinuum’s H2-1. The framework supports fair cross-platform benchmarking and offers a pathway toward hybridized and fault-tolerance–oriented metrics as quantum devices mature.

Abstract

As quantum computing advances toward the late-NISQ and early fault-tolerant eras, scalable and platform-independent benchmarks are essential for quantifying computational capacity in a classically verifiable manner. We introduce two volumetric benchmarks, Clifford Volume and Free Fermion Volume, that assess quantum hardware by testing the execution of random Clifford and free fermion operations. These two groups of unitaries possess a combination of properties that make them ideal for benchmarking: (i) each is individually efficient to simulate classically, enabling verification at scale; (ii) together they form a universal gate set; (iii) they serve as essential algorithmic primitives in practical applications (including shadow tomography and quantum chemistry); and (iv) their definitions are formulated abstractly, without explicit reference to hardware-specific features such as qubit connectivity or native gate sets. This framework thus enables scalable and fair cross-platform comparisons and tracks meaningful computational advancement. We demonstrate the practical feasibility of these benchmarks through extensive numerical simulations across realistic noise parameters and through experimental validation on Quantinuum's H2-1 trapped-ion quantum computer, which achieves a Clifford Volume of 34.

Figures (16)

• Figure 1: Graphical representation of the $n$-qubit step in the Clifford Volume benchmark protocol. The quantum circuit is initialized in the $\left|{0}\right\rangle^{\otimes n}$ state. Then, a Clifford unitary is applied, which is decomposed and compiled according to the characteristics of the target platform. The circuit is run for a prescribed number of times (see Appendix \ref{['appendix : CV_step_by_step']}) in order to evaluate the expectation values of selected $n$-qubit Pauli operators $P_i \in \{I, X, Y, Z\}$, grouped into subsets $S_i \in \mathcal{S}$ and $D_i \in \mathcal{D}$, corresponding to stabilizer and destabilizer sets, respectively.
• Figure 2: Distribution of simulated expectation values for randomly selected Pauli operators in the CLV benchmark, with readout error $p_m = 10^{-2}$ and two-qubit gate error $p_{2Q} = 10^{-3}$. Subfigures (a)–(d) show histograms of the expectation values $\langle P_i \rangle$ for stabilizer (blue) and destabilizer (red) operators at different qubit counts: $n = 5$, $15$, $25$, and $35$, respectively. The horizontal dashed lines indicate the threshold values: $\frac{1}{e}$ for stabilizers and $\frac{1}{2e}$ for destabilizers. Subfigure (e) shows the average expectation values of stabilizers (blue) and destabilizers (red) as a function of the number of qubits. Shaded regions indicate the standard deviation over the ensemble, while dashed lines represent the maximum deviation from the ideal value for each ensemble. Vertical dashed lines correspond to the qubit counts shown in subfigures (a)–(d).
• Figure 3: The simulated minimum stabilizer expectation values and the maximum destabilizer absolute expectation values as a function of the number of qubits in the CLV benchmark . Expectation values are estimated by simulating $4096$ measurement shots per observable. For each qubit number $n$, four randomly sampled $n$-qubit Clifford unitaries are implemented. The blue curve shows the smallest measured expectation value among all sampled stabilizer operators, while the red curve shows the largest absolute value of the measured expectation values of destabilizer operators (i.e., the one farthest from zero). Shaded regions indicate the $\pm 2\sigma$ statistical uncertainty used in the benchmark threshold criteria. Horizontal dashed lines mark the acceptance thresholds $\tfrac{1}{e}$ for stabilizers and $\tfrac{1}{2e}$ for destabilizers.
• Figure 4: CLV benchmark scores for each pair of error parameters. The horizontal axis corresponds to the two-qubit gate error probability $p_{2Q}$, and the vertical axis to the readout error probability $p_m$. Each cell indicates the largest qubit count $n$ for which the benchmark criteria are satisfied under the corresponding noise configuration.
• Figure 5: Schematic representation of the Free-Fermion Volume benchmark protocol. The circuit begins with all qubits initialized in the $|0\rangle$ state, except for a randomly chosen qubit prepared in the $|+\rangle$ state. This preparation distinguishes the corresponding Majorana operator, which -- unlike all others -- has an expectation value of $1$ in this state. A randomly sampled free-fermion unitary is then applied, decomposed into a brick-wall pattern of single- and two-qubit gates derived from Givens rotations of the corresponding $SO(2n)$ orthogonal matrix. After the unitary, single-qubit measurements are performed to evaluate the expectation values of selected Majorana mode operators (mapped to Pauli operators on qubits) to assess the precision of the implementation and verify the orthogonality conditions of the underlying $SO(2n)$ transformation(see Appendix \ref{['FFV_step_by_step']}).