Cutting Quantum Circuits Beyond Qubits

We extend quantum circuit cutting to heterogeneous registers comprising mixed-dimensional qudits. By decomposing non-local interactions into tensor products of local generalised Gell-Mann matrices, we enable the simulation and execution of high-dimensional circuits on disconnected hardware fragments. We validate this framework on qubit–qutrit ($2$–$3$) interfaces, achieving exact state reconstruction with a Total Variation Distance of 0 within single-precision floating-point tolerance. Furthermore, we demonstrate the memory advantage in an 8-particle, dimension-8 system, reducing memory usage from 128 MB to 64 KB per circuit.

💡 Research Summary

The paper presents a comprehensive extension of quantum circuit cutting to heterogeneous registers composed of mixed‑dimensional qudits, such as qubit–qutrit (2‑3) interfaces. Traditional circuit‑cutting methods rely on the Pauli basis and are limited to two‑level systems; this work replaces that basis with the generalized Gell‑Mann matrices, which form an orthogonal operator basis for any dimension d. By expressing non‑local two‑qudit gates (in particular the generalized CX/CSUM gate) as a linear combination of tensor products of local Gell‑Mann operators, the authors derive an explicit decomposition formula (Eq. 2.10) that works for arbitrary dimensions d₁ ≠ d₂.

The decomposition proceeds by constructing local bases (B_1) and (B_2) that contain the identity and the appropriate Gell‑Mann matrices, then expanding the projectors (P_r) and shift operators (X_r) in these bases. Coefficients are obtained via trace inner products, yielding a set of weighted local operators ({A_i\otimes B_i}). The number of terms in the full expansion scales as (d_1(d_1 d_2)^2), but many coefficients are zero or negligible in practice.

After cutting a circuit, each fragment is executed independently on a small quantum processor or simulator. The authors introduce a mixed‑radix reconstruction algorithm (Algorithm 1) that maps the amplitude vector of each fragment back to the logical basis of the original circuit, handling asymmetric dimension vectors and required permutations. The algorithm iterates over all amplitude indices, decomposes each index into its mixed‑radix representation according to the ordered list of subsystem dimensions, permutes to logical order, and assigns probabilities as the squared magnitude of the corresponding amplitude.

Experimental validation is performed on both homogeneous (qubit–qubit) and heterogeneous (qubit–qutrit) cuts. Probability distributions reconstructed from the cut fragments match the original distributions with a total variation distance (TVD) of 0.00000 up to the 12th decimal place, confirming exact reconstruction within single‑precision floating‑point limits.

A key contribution is the demonstration of memory savings. Simulating an eight‑qudit system where each qudit has dimension 8 requires (8^8) complex amplitudes, i.e., 128 MB in Complex64 precision. By cutting the circuit into two four‑qudit subcircuits, each fragment needs only (8^4) amplitudes, reducing memory per fragment to 64 KB—a 2000‑fold reduction. The authors benchmarked this on a 2022 Apple M2 laptop, showing that with a 150 MB memory cap the full simulation fails, while the cut simulation completes (albeit with a ten‑fold increase in wall‑clock time due to the 532 fragment evaluations).

To mitigate the explosion of terms in the full basis expansion, the paper explores Schmidt (singular‑value) decomposition of the two‑qudit gate. By performing an SVD on the unitary, the gate can be expressed with only (\min(d_1,d_2)) terms, dramatically reducing both memory and runtime. For a 12‑12 CX gate, the full basis would need 225 terms, whereas the Schmidt method uses only 12, cutting memory from ~8 MB to 1 MB and achieving speedups of 5× or more for moderate dimensions.

Scaling analyses (Figures 1‑3) illustrate how TVD remains negligible when truncating coefficients at (10^{-2}) even for systems with effective dimension up to (10^6), while runtime savings increase with system size. Beyond (10^9) dimensions, accumulated truncation error and floating‑point limits cause TVD growth, indicating a practical bound for aggressive truncation.

In conclusion, the authors provide (1) a mathematically rigorous, dimension‑agnostic circuit‑cutting framework based on generalized Gell‑Mann matrices, (2) an efficient mixed‑radix reconstruction algorithm for heterogeneous cuts, (3) empirical evidence of exact state reconstruction and massive memory reduction, and (4) a Schmidt‑decomposition‑based optimization that minimizes the number of tensor‑product terms. This work opens the door to distributed quantum computing across heterogeneous quantum hardware, enabling larger algorithms to be run on modest NISQ devices while managing memory constraints and preserving high fidelity.