Classical simulation of quantum circuits using a multiqubit Bloch vector representation of density matrices

In the Bloch sphere picture, one finds the coefficients for expanding a single-qubit density operator in terms of the identity and Pauli matrices. A generalization to $n$ qubits via tensor products represents a density operator by a real vector of length $4^n$, conceptually similar to a state vector. Here, we study this approach for the purpose of quantum circuit simulation, including noise processes. The tensor structure leads to computationally efficient algorithms for applying circuit gates and performing few-qubit quantum operations. In view of variational circuit optimization, we study “backpropagation” through a quantum circuit and gradient computation based on this representation, and generalize our analysis to the Lindblad equation for modeling the (nonunitary) time evolution of a density operator.

Figure 1

Schematic excerpt of a parametrized quantum circuit, with intermediate states $\rho$ and ${\rho }^{\prime }$. The blue boxes represent unitary circuit gates.

Figure 2

A single parametrized layer of the circuit model.

Figure 3

Variation of (a) fidelity, (b) loss, and (c) trace distance after 500 optimization iterations over $\beta$ values from $\beta =0$ to $\beta =20$ at 0.1 intervals for $H_{\text{1D}}$ in (51) with $n=4$, with chosen parameters $J=-1, g=0.3, h=0.2$. The shaded region indicates the $95\%$ confidence interval.

Figure 4

Convergence of the VQT algorithm optimization procedure at $\beta =0.5$ for the Heisenberg Hamiltonian (52) on a $2\times 2$ lattice with parameters $J_h=1, J_v=0.6$. The shaded regions indicate $95\%$ confidence intervals.

Figure 5

Metrics for approximating the target thermal density ${\sigma }_{\beta }$ in (44) as a function of $\beta$, with 500 optimization iterations using the VQT algorithm with parameters as in Fig. 4. The shaded regions indicate $95\%$ confidence intervals.

Figure 6

Benchmark comparison of our Bloch vector methodology (blue) with a conventional implementation by matrix conjugations (orange), for the task of applying typical single- and two-qubit gates.