Circuit connectivity boosts by quantum-classical-quantum interfaces

High-connectivity circuits are a major roadblock for current quantum hardware. We propose a hybrid classical-quantum algorithm to simulate such circuits without swap-gate ladders. As the main technical tool, we introduce quantum-classical-quantum interfaces. These replace an experimentally problematic gate (e.g., a long-range one) with single-qubit random measurements followed by state preparations sampled according to a classical quasiprobability simulation of the noiseless gate. Each interface introduces a multiplicative statistical overhead which, remarkably, is independent of the on-chip qubit distance. Hence, by applying interfaces to the longest-range gates in a target circuit, significant reductions in circuit depth and gate infidelity can be attained. We numerically show the efficacy of our method for a Bell-state circuit for two increasingly distant qubits and a variational ground-state solver for the transverse-field Ising model on a ring. Our findings provide a versatile toolbox for error-mitigation and circuit boosts tailored for noisy, intermediate-scale quantum computation.

Figure 1

Schematics of our method. (a) A QCQ interface $\mathcal{V}(\mathit{a},\mathit{b})$ applying the identity operator $U_k=\mathbb{1}$ between qubits 1 and $N$. We measure the POVM $\mathit{M}$ on both qubits reprepare them in a product state that depends on the simulated gate and the outcome $\mathit{a}$. The other $N-2$ qubits are left untouched. (b) An exemplary four-qubit circuit (left) is simulated by a hybrid quantum-classical circuit (right), where the non-nearest-neighbor gates ${\mathcal{U}}_1$ and ${\mathcal{U}}_3$ are substituted by QCQ interfaces $[{\mathcal{V}}_1({\mathit{a}}_{s_1},{\mathit{b}}_{s_1})$ and ${\mathcal{V}}_3({\mathit{a}}_{s_3},{\mathit{b}}_{s_3})$, respectively]. The summation over $({\mathit{a}}_{s_1},{\mathit{b}}_{s_1},{\mathit{a}}_{s_3},{\mathit{b}}_{s_3})$ represents the average over all interface outcomes sampled (see text).

Figure 2

CHSH violation as a function of the number of qubits. These results were obtained with an LPDO simulation where $D=12$ and $\kappa =24$. In addition to the gate noise, we apply a depolarizing channel to simulate measurement noise with ${\lambda }_{\text{meas}}=0.01$ and repreparation noise with ${\lambda }_{\text{reprep}}=0.005$. The classical bound (pink) and maximal violation orange) are 2 and $2\sqrt{2}$, respectively, for all $d$. We see that the violation in the noisy circuit (green) decreases linearly with the number of qubits as a result of the $4(d-2)+1$ noisy swap gates required to prepare the state. Our algorithm provides the maximum CHSH violation up to statistical fluctuations independent of the distance between the qubits. This comes at a cost of sampling $M=60000$ measurement-and-repreparation steps to estimate the violation.

Figure 3

The Hamiltonian variational ansatz circuit for the ground state of the TFIM. The parameters $\{{\beta }_i,{\gamma }_i\}$ for $i=1,...,p$ can be found with a variational quantum eigensolver optimization [9]. Each layer in the circuit contains a long-range two-qubit ZZ rotation. We assume that the distance between the first and last qubit is $N-2$. Implementing the nearest-neighbor ZZ gates comes at a cost of $2(N-1)$ cnot gates. The long-range ZZ rotations require $4(N-2)+1$ cnot gates since we must use a swap chain to bring the first and last qubit together. The total number of cnot gates per layer is therefore dominated by the implementation of long-range ZZ rotations.

Figure 4

Comparison of QCQ interface simulation with both noisy and noiseless TFIM circuits for (a) $N=4$ and (b) $N=8$ qubits obtained with a full density matrix simulation. Each dot represents the average energy $\mathbb{E}\left[\langle {\widehat{H}}_M\rangle \right]$ estimated over 50 separate instances. The error bars indicate the standard deviation. As the number of samples $M_s$ increases, the statistical fluctuations of our method become small in accordance with the central limit theorem. We can determine the scaling of the size of the error bars by fitting $\sigma =\overline{\sigma }/\sqrt{N_{\text{samples}}}$. While for four qubits $\overline{\sigma }\approx 27.8$, for eight qubits we have $\overline{\sigma }\approx 76.5$. This scaling only depends on the mean negativity, which differs between the two circuits because we apply a different ZZ rotation on each circuit. The energy of the noiseless circuit (orange dashed line) corresponds to the ground-state energy $E_{\text{gs}}$. The noisy circuit (green dashed line) shows the energy obtained when we apply depolarizing channels with ${\lambda }_{\text{unit}}=0.005$ to the cnot gates in the circuit. We see that for both four and eight qubits, our algorithm provides a significant improvement on the final estimated energy of the circuit for a reasonable number of measure-and-reprepare steps. In (b) we observe that the large number of number of noisy cnot gates dominates the simulation; hence the improvement is not as significant as for four qubits.

Figure 5

Comparison of double-QCQ-interface simulation with both noisy and noiseless TFIM circuits for (a) $N=4$ and (b) $N=8$ qubits. These results were obtained with a full density matrix simulation. In (a), we see that we can almost approximate the true ground-state energy of the four-qubit state, because the only noisy operations are the 12 cnot gates required for implementing the six nearest-neighbor ZZ gates in layers 1 and 2. In (b) we see a more significant improvement over the energies from Fig. 4, but still the noise dominates. Since we apply the QCQ method twice, the standard deviation $\sigma =\overline{\sigma }/\sqrt{N_{\text{samples}}}$ of the error bars increases quadratically, as per Eq. (6). We find $\overline{\sigma }\approx 333.1$ and $\overline{\sigma }\approx 856.8$ for four and eight qubits, respectively.

Figure 6

Comparison of a QCQ interface simulation with both noisy and noiseless circuits for a 20-qubit TFIM circuit. These results were obtained with an LPDO simulation where $D=50$ and $\kappa =50$. Only two of the eight layers of the circuit are simulated here, to keep simulation errors under control. The sample variance $\widehat{\sigma }\approx 195.0$.

Figure 7

Illustration of the lower bound of Eq. (F8). The circuit consists of an initial state ${|+\rangle }^{\otimes 4}$ to which we apply a varying number of cnot gates with random control and target qubits. We set the noise to $\lambda =0.005$ and take $D=4$ and $\kappa =16$. The red line indicates the true accuracy of the LPDO simulation by comparing it with the exact full density matrix simulation. The orange line gives the runtime fidelity estimate. We see that the accuracy of the simulation degrades as we add more two-qubit gates and depolarizing channels. The runtime fidelity gives an estimate two orders of magnitude above the exact error, indicating that for this example, the bound is a conservative estimate of the simulation error.

Figure 8

Monte Carlo random walk for interface negativity optimization of the ZZ gate used in Sec. 4b of the main text. The total number of steps in the annealing schedule is 5000. The gray dashed line indicates the mean average squared negativity of the pseudoinverse, whereas the blue line indicates the one for the newly accepted $\mathfrak{T}$'s during the Monte Carlo random walk. The inset shows the adaptive scheme that fine-tunes the search with the temperature and variance, given in black and red, respectively.

Figure 9

Improvement of energy-estimator variance for the eight-qubit TFIM circuit experiment of Fig. 4 in the main text. Sample variance is estimated over 50 runs. The red line shows the sample variance corresponding to the canonical dual frame given by the pseudoinverse of $T$. In blue we see the variance of the energy corresponding to the dual frame obtained from the Monte Carlo search.