Cameron Calcluth

We examine the ability of gate-based continuous-variable quantum computers to outperform qubit or discrete-variable quantum computers. Gate-based continuous-variable operations refer to operations constructed using a polynomial sequence of elementary gates from a specific finite set, i.e., those selected from the set of Gaussian operations and cubic phase gates. Our results show that for a fixed energy of the system, there is no superpolynomial computational advantage in using gate-based continuous-variable quantum computers over discrete-variable ones. The proof of this result consists of defining a framework–––of independent interest---that maps quantum circuits between the paradigms of continuous- and discrete-variables. This framework allows us to conclude that a realistic gate-based model of continuous-variable quantum computers, consisting of states and operations that have a total energy that is polynomial in the number of modes, can be simulated efficiently using discrete-variable devices. We utilize the stabilizer subsystem decomposition [Shaw et al., PRX Quantum 5, 010331] to map continuous-variable states to discrete-variable counterparts, which allows us to find the error of approximating continuous-variable quantum computers with discrete-variable ones in terms of the energy of the continuous-variable system and the dimension of the corresponding encoding qudits.

Classically simulating circuits with bosonic codes is challenging due to the prohibitive cost of simulating quantum systems with many, possibly infinite, energy levels. We propose an algorithm to simulate circuits with encoded Gottesman-Kitaev-Preskill (GKP) states, specifically for odd-dimensional encoded qudits. Our approach is tailored to be especially effective in the most challenging but practically relevant regime, where the codeword states exhibit high (but finite) squeezing. Our algorithm leverages the Zak-Gross Wigner function introduced by J. Davis et al. [arXiv:2407.18394], which represents infinitely squeezed encoded stabilizer states positively. The runtime of the algorithm scales with the negativity of the Wigner function, allowing for efficient simulation of certain large-scale circuits — namely, input stabilizer GKP states undergoing generalized GKP-encoded Clifford operations followed by modular measurements — with a high degree of squeezing. For stabilizer GKP states exhibiting 12 dB of squeezing, our algorithm can simulate circuits with up to 1,000 modes with less than double the number of samples required for a single input mode, in stark contrast to existing simulators. Therefore this approach holds significant potential for benchmarking early implementations of quantum computing architectures utilizing bosonic codes.