GCAMPS: A Scalable Classical Simulator for Qudit Systems

TL;DR

This work generalises the CAMPS method to higher quantum degrees of freedom — qudit simulation, resulting in a generalised CAMPS (GCAMPS), and shows that similar to the case of qubits, qutrit systems also benefit from a comparable speedup using these techniques.

Abstract

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.

Figures (7)

• Figure 1: The qudit stabilizer states correspond to positions on a $d$-digit clock, where the Pauli $X$ operator moves the clock's hand. For (a) $d=2$ this corresponds to up and down spins. (b) When $d \geq 3$ there are more possible states. Similarly, $Z$ and $Y$ rotate between $d$ orthogonal basis states.
• Figure 2: (a) A tensor network representation of the product $AB$, i.e. matrix multiplication. (b) The graphical representation of the MPS in Equation \ref{['eq:mps']}
• Figure 3: GCAMPS Workflow: The state is represented by an MPS $\ket{MPS}$ and a leading Clifford operation $C$ represented by a stabilizer tableau. (Top loop) Clifford operations directly update the tableau resulting in the top loop. (Bottom loop) Non-Clifford operations must first be decomposed into a sum of Paulis $\sum_i P_i$ and commuted through the tableau. The commuted operator is the applied directly to the MPS, after which the MPS may be optimised by extracting entanglement using Clifford operations that are applied to the stabilizer tableau. GCAMPS extends CAMPS by incorporating a qudit stabilizer simulator for the Clifford $C$ and extending the decomposition and optimisation steps in the bottom loop to utilise the qudit Cliffords.
• Figure 4: The $T$-doped random Clifford circuits simulated. Each layer consists of a random Clifford operation, followed by a $T$ gate on the first qubit. This is repeated $t$ times, with the circuit depth being the number of layers, which is equal to $t$, the number of non-Clifford $T$ gates in the circuit.
• Figure 5: (a) The scaled bond-dimension after each layer for of both (i) qubit and (ii) qutrit $T$-doped random Clifford circuits simulated with GCAMPS as a function of the scaled number of Clifford + T layers $t$. The bond dimension can be though of in analogy with the memory cost associated with the MPS as discussed in Section \ref{['sec:mem']}. We note a similar scaling behaviour for both qubits and qutrits, with the qubit simulation exhibiting a phase transition at a lower circuit depth as discussed in Section \ref{['sec:results']}. Note that due to computational constraints the $N=16, 18$ qutrit simulations were only simulated up to $23$ layers. (b) Conventional MPS simulation for up to $N=12$, we note that the bond dimension of the MPS saturates almost immediately after very few layers.