GCAMPS: A Scalable Classical Simulator for Qudit Systems

Classical simulations of quantum systems are notoriously difficult computational problems, with conventional state vector and tensor network methods restricted to quantum systems that feature only a small number of qudits. The recently introduced Clifford Augmented Matrix Product State (CAMPS) method offer scalability and efficiency by combining both tensor network and stabilizer simulation techniques and leveraging their complementary advantages. This hybrid simulation method has indeed demonstrated significant improvements in simulation performance for qubit circuits. Our work generalises the CAMPS method to higher quantum degrees of freedom – qudit simulation, resulting in a generalised CAMPS (GCAMPS). Benchmarking this extended simulator on quantum systems with three degrees of freedom, i.e. qutrits, we show that similar to the case of qubits, qutrit systems also benefit from a comparable speedup using these techniques. Indeed, we see a greater improvement with qutrit simulation compared to qubit simulation on the same $T$-doped random Clifford benchmarking circuit as a result of the increased difficulty of conventional qutrit simulation using tensor networks. This extension allows for the classical simulation of problems that were previously intractable without access to a quantum device and will open new avenues to study complex many-body physics and to develop efficient methods for quantum information processing.

💡 Research Summary

The paper introduces GCAMPS, a generalisation of the recently proposed Clifford‑augmented Matrix Product State (CAMPS) technique, to enable efficient classical simulation of qudit (d‑level) quantum systems. Classical simulation of quantum circuits is notoriously hard because the state‑vector size grows as O(dⁿ) and tensor‑network methods such as Matrix Product States (MPS) become costly when the entanglement or local dimension d is large. Stabiliser‑based simulators, on the other hand, are efficient for Clifford circuits but scale exponentially when non‑Clifford gates (e.g., T‑gates) are present. CAMPS combined these two approaches for qubits, achieving speed‑ups by using stabiliser updates for Clifford parts and MPS for the remaining entangled state, while periodically applying “disentanglers” (Clifford operators) to keep the MPS bond dimension low.

GCAMPS extends this hybrid framework to arbitrary d. The authors first generalise the Pauli group: X and Z act as cyclic shift and phase operators with ω = exp(2πi/d), and Y = XZ (or τXZ for even d). The stabiliser tableau becomes a 2n × (2n + 1) matrix with entries in ℤ_d, storing exponents of X and Z and a phase column in powers of ω. Clifford gates for qudits (generalised Hadamard H, phase S, and the SUM entangling gate) map Pauli strings to Pauli strings, allowing tableau updates via simple modular arithmetic. Non‑Clifford gates are introduced through a d‑dimensional T‑gate (√S) or, more conveniently for simulation, a parametrised R_z(θ) rotation; universality follows from the Clifford + T set.

The GCAMPS algorithm proceeds gate‑by‑gate. For a Clifford segment, the tableau is updated directly. When a non‑Clifford gate is encountered, it is applied to the MPS, after which the algorithm searches for a suitable Clifford “disentangler” that, when conjugated with the current state, reduces the bond dimension. Finding such a disentangler requires expressing an arbitrary Pauli string as a product of stabiliser and destabiliser generators, which is solved by Gaussian elimination over ℤ_d. The authors note that this search is efficient for d = 3 (qutrits) but lacks a polynomial‑time method for larger d, limiting the practical scope of GCAMPS to low‑dimensional qudits.

Benchmarking focuses on random Clifford circuits doped with T‑gates on qutrit systems. The authors vary the number of T‑gates relative to the number of qudits n and compare GCAMPS against a pure MPS simulator. Results show that when the T‑gate count is less than about 0.5 n, GCAMPS reduces memory consumption by a factor of 3–5 and runtime by 2–4 relative to MPS. The speed‑up is more pronounced for qutrits than for qubits because the tensor‑network cost grows faster with d, while the stabiliser component’s cost remains essentially independent of d. When the T‑gate count exceeds the number of qudits, the advantage disappears and GCAMPS reverts to MPS‑like scaling.

The discussion acknowledges limitations: (i) the need for an efficient, scalable disentangler‑search algorithm for arbitrary d, (ii) reduced benefits for circuits with many non‑Clifford gates, and (iii) the current focus on regimes where the number of T‑gates is smaller than the system size. Future work is suggested in developing better disentangler discovery methods, extending benchmarks to d > 3, and integrating machine‑learning‑driven optimisation for automatic selection of Clifford transformations.

In summary, GCAMPS provides a practical hybrid simulation framework for higher‑dimensional quantum systems, offering substantial memory and time savings for low‑magic, low‑entanglement qudit circuits. It opens the door to classical studies of many‑body physics and quantum information protocols that were previously out of reach for classical computers, especially in the emerging area of qutrit‑based quantum technologies.