Simulating the Open System Dynamics of Multiple Exchange-Only Qubits using Subspace Monte Carlo

Abstract

We propose a Monte Carlo based method for simulating the open system dynamics of multiple exchange-only (EO) qubits. In the EO encoding, the total spin projection quantum number along the $z$-axis of the three constituent spins remains unchanged under exchange operations, in contrast to the open system (or multi-qubit miscalibration) setting where coherent and incoherent mixing of states with different quantum numbers occurs. In our approach, we choose to measure the total spin component along the $z$-axis of each EO qubit after every logical quantum operation, which decoheres coherent mixtures of states with different spin projection quantum numbers. Independent simulations thus give different trajectories of the system in the associated subspaces, so we refer to this method as the Subspace Monte Carlo method. With each EO qubit having a definite spin projection quantum number, the density matrix of $n$ qubits can be represented by a vector of dimension $3^{2n}$, instead of $8^{2n}$, with an additional vector of dimension $n$ to label the quantum number of each qubit. We show that this approximation of the dynamics remains faithful to the true dynamics when the simulated circuits twirl the noise, converting coherent errors to stochastic errors, which can be achieved using randomized compiling. We use this simulation approach to study how correlations in measurement outcomes of circuits with reset-if-leaked gadgets, such as a multi-round Bell state stabilization circuit that uses 6 EO qubits, are affected by the choice of CNOT implementations.

Figures (13)

• Figure 1: Illustration of the Subspace MC approximation on two EO qubits. After the application of the logical quantum operation $\mathcal{U} \cdot \mathcal{E}$, a measurement of the $\hat{S}_z$ operator on each qubit is performed. This projects each qubit onto a subspace with a fixed $s_z$ value.
• Figure 2: Repeated SWAP pairs. (a) Illustration of the quantum circuit implementing $m$ repeated applications of a pair of SWAPs. The circuit is terminated by a computational basis measurement of each qubit. (b) Same as in (a), except we include randomized compiling for the pair of SWAP gates. (c) and (d) Probability of measuring the outcome 0 for both qubits when simulating the circuit in (a) and (b) respectively when acting on the initial Bell state $\left| \Phi \right \rangle = \frac{1}{\sqrt{2}} \left( \left| 00 \right \rangle + \left| 11 \right \rangle \right)$. Simulation results are averaged over 100 noise realizations using noise model NM1, with the error bars being the $2\sigma$ confidence intervals generated by performing a bootstrap over the 100 noise realizations. For the Subspace MC simulations, 50 MC trials are performed per noise realization.
• Figure 3: Two-qubit RB. Average survival probability as a function of circuit depth $L$ for 2-qubit randomized benchmarking using the LCCX gate. At every depth, 50 random circuits are averaged over. The two simulation methods presented use the same 10 independent noise realizations using noise model NM1. For the subspace MC simulations, 5 Monte Carlo trials are performed per noise realizations. Error bars are $2\sigma$ confidence intervals generated by performing a bootstrap over the 10 noise realizations.
• Figure 4: RiL gadget benchmarking protocol. A depth $L$ 1-qubit RB circuit with RiL gates interleaved. Each RiL gate is chosen such that the reset state matches the ideal state at that point in the circuit.
• Figure 5: Simulation of the RiL gadget benchmarking protocol. Average survival probability as a function of circuit depth $L$ for the RiL gadget benchmarking protocol illustrated in Fig. \ref{['fig:RBRiL']}. At every depth, 25 random circuits are averaged over. The two simulation methods presented use the same 100 independent noise realizations. For the subspace MC simulations, 10 Monte Carlo trials are performed per noise realization. Error bars are $2\sigma$ confidence intervals generated by performing a bootstrap over the 100 noise realizations using noise model NM1. The inset shows per-Clifford leakage probability as a function of circuit depth $L$. As expected, the leakage probability approaches a constant, corresponding to an asymptotic per-Clifford leakage probability of $6.1\times 10^{-3} \pm 2 \times 10^{-4}$ in this example.