Digital-analog quantum computation with arbitrary two-body Hamiltonians

Digital-analog quantum computing is a computational paradigm which employs an analog Hamiltonian resource together with single-qubit gates to reach universality. Here, we design a new scheme which employs an arbitrary two-body source Hamiltonian, extending the experimental applicability of this computational paradigm to most quantum platforms. We show that the simulation of an arbitrary two-body target Hamiltonian of $n$ qubits requires $\mathcal{O}\left(n^2\right)$ analog blocks with guaranteed positive times, providing a polynomial advantage compared to the previous scheme. Additionally, we propose a classical strategy which combines a Bayesian optimization with a gradient descent method, improving the performance by $\sim 55\%$ for small systems measured in the Frobenius norm.

Figure 1

sDAQC circuit for the Ising $ZZ$ ATA Hamiltonian for five qubits. For simulating an arbitrary evolution, we sandwich several analog blocks with a couple of $X$ gates applied to all combinations of two qubits. The blocks labeled with $t_{i,j}$ represent the unitary evolution under the source Hamiltonian for the time $t_{i,j}$; this is $e^{-i{t}_{i,j}{H}_{\mathrm{S}}}$.

Figure 2

sDAQC circuit for an arbitrary ATA Hamiltonian for five qubits. The protocol for simulating any target Hamiltonian can be divided into two steps. In the first step (top), we select each of the $n(n-1)/2$ pairs of qubits in our system. In the second step (bottom), for each pair of qubits, we apply all the nine possible combinations of the Pauli gates. The number of analog blocks for this protocol scales quadratically with the number of qubits, $\mathcal{O}\left(n^2\right)$.

Figure 3

Frobenius distance between the optimized circuit and the exact evolution vs the number of digital-analog blocks. (a) We solve the problem of the nearest-neighbor Hamiltonian for six qubits. We show two cases, one in which the couplings in the system Hamiltonian is homogeneous, and the other where the coupling strengths are in a normal distribution around the values of the former. We also distinguish two versions for the optimization process, which fixes the length of the analog blocks or leaves them as a variables. As a baseline, we represent the distances obtained with first-order Trotterization up to $n_T=19$ Trotter steps. The shown values indicate the mean Frobenius distance and the error bars show the interquartile range. The inset on top provides an extended view in the coordinate axis for a higher number of blocks in logarithmic scale. (b) We repeat the procedure for a single run with a different classical simulation method, in which we employ a first-order Trotter approximation for computing both the circuit and the target.