Mitigating Realistic Noise in Practical Noisy Intermediate-Scale Quantum Devices

Quantum error mitigation (QEM) is vital for noisy intermediate-scale quantum (NISQ) devices. While most conventional QEM schemes assume discrete gate-based circuits with noise appearing either before or after each gate, the assumptions are inappropriate for describing realistic noise that may have strong gate dependence and complicated nonlocal effects, and general computing models such as analog quantum simulators. To address these challenges, we first extend the scenario, where each computation process, being either digital or analog, is described by a continuous time evolution. For noise from imperfections of the engineered Hamiltonian or additional noise operators, we show it can be effectively suppressed by a stochastic QEM method. Since our method assumes only accurate single qubit controls, it is applicable to all digital quantum computers and various analog simulators. Meanwhile, errors in the mitigation procedure can be suppressed by leveraging the Richardson extrapolation method. As we numerically test our method with various Hamiltonians under energy relaxation and dephasing noise and digital quantum circuits with additional two-qubit crosstalk, we show an improvement of simulation accuracy by 2 orders. We assess the resource cost of our scheme and conclude the feasibility of accurate quantum computing with NISQ devices.

Figure 1

(a) Continuous QEM. With discretized time step $\delta t$, each recovery operation is weakly and “continuously” acted after each noisy evolution of time $\delta t$. Here different colors represent different recovery operations. The output state is measured and repeated to obtain $N_s$ outcomes $\{O_m\}$, and their average corresponds to the error-mitigated outcome. (b) Stochastic QEM. We can equivalently realize (a) by $\delta t\to 0^{+}$ and randomly applying a small number $n_m$ of strong recovery operations as in Algorithm 1. The time $\{t_{\mathrm{jp}}^{m,k}{\}}_m$ to apply recovery operations of the $m\mathrm{th}$ run are predetermined, which can be further preengineered into the original evolution via a noisy time evolution of a modified Hamiltonian.

Figure 2

(a) Schematic diagram of the error-mitigated AQS or DQS with controllable single-qubit operations. The AQS or DQS is realized by a continuous process under the ideal driving Hamiltonian. We denote the noisy unitary and stochastic processes by $\mathcal{H}$ and $\mathcal{L}$ with the superoperator formalism. Our work considers joint dynamics of all qubits sandwiched with a small number [$\mathcal{O}(1)$] of preengineered single-qubit dynamics to mitigate the errors accumulated in the evolution, which can also be regarded as a modified evolution ${\mathcal{H}}^{\prime }$. (b) Two schemes of the error-mitigated process or gate with modified pulse sequences. Dashed lines represent the original pulse that constructs the target process or multiqubit gates. Provided controllable drive that could be freely turned on and off, we can synthesize the error-mitigated process and gate by modifying the original pulse sequence as shown in scheme I, which corresponds to (a). In the case of restricted driving operations, we can alternatively apply a strong and fast single-qubit pulse to the original pulse to mitigate either process errors or gate errors as shown in scheme II. We note that scheme II could similarly be applied in (a), which introduces a negligible error of $\mathcal{O}(\delta t^3)$ when each single-qubit gate is implemented in $\delta t\ll T$.

Figure 3

Numerical test of the QEM schemes without [(a)–(c)] and with $10\mathrm{\%}$ model estimation error ${\lambda }_{\exp }=1.1{\lambda }_{\mathrm{est}}$ [(d)–(g)]. We consider the dynamics of two-dimensional (2D) anisotropic Heisenberg Hamiltonian with energy relaxation and dephasing noise. (a)–(f) consider a four-qubit Hamiltonian with a finite $({10}^6)$ number of samples. (a),(d) compare the time-evolved nearest-neighbor correlation function $\sum _{⟨ij⟩}{\sigma }_x^{(i)}{\sigma }_x^{(j)}/4$. (b),(e) shows the error between the exact value and the error-mitigated value. (c),(f) show the fidelity of the effective density matrix ${\rho }_{\mathrm{eff}}^{\alpha }$ and the ideal one ${\rho }_I$ under different error-mitigation scheme $\alpha$. (g) considers an eight-qubit Hamiltonian with an infinite number of samples. The hybrid error-mitigation scheme suppresses the error up to about 4 orders of magnitude even with $10\mathrm{\%}$ model estimation error.

Figure 4

Stochastic QEM for eight-qubit superconducting quantum circuits with environmental and crosstalk noise. (a) considers a $d$-depth parameterized quantum circuit with single-qubit rotations $R_{\alpha }$ ($\alpha \in \{X,Y,Z\}$) and cnot gates. The rotation angles are randomly generated from $[0,2\pi ]$. (b) shows the realization of the cnot gate via the cr gate $U_{\mathrm{cr}}=\exp [i\pi {\sigma }_z^{(c)}{\sigma }_x^{(t)}/4]$ and single-qubit gates $R_z^{\pi /2}$ and $R_x^{\pi /2}$ up to a global phase $e^{i\pi /4}$. (c) shows the fidelity dependence of circuit depth $d$ with and without QEM.

Figure 5

(a) Cost versus total noise strength $\mathrm{\Lambda }=NT({\lambda }_1+{\lambda }_2)$. We consider a general $N$-qubit Hamiltonian $H_{\mathrm{sim}}$ with single-qubit energy relaxation (${\lambda }_1$) and dephasing noise (${\lambda }_2={\lambda }_1$), and evolution time $T$. The inset shows the cost versus a different number of qubits with $({\lambda }_1+{\lambda }_2)T=0.01$. (b) Simulation accuracy $\epsilon \propto C/\sqrt{N_s}$ with different number of samples $N_s$ with $T=1\mu \mathrm{s}$, ${\lambda }_1+{\lambda }_2=0.01$ MHz, $N=100$ (red), and $N=50$ (blue). We consider only pessimistic estimation $C/\sqrt{N_s}$ and the error $\epsilon$ can be much smaller in practice.

Figure 6

Numerical test of the performance of error-mitigation schemes for transverse-field Ising model without model estimation error [(a)–(c)] and with $10\mathrm{\%}$ model estimation error ${\lambda }_{\exp }=1.1{\lambda }_{\mathrm{est}}$ [(d)–(g)]. We consider time evolution of the Ising Hamiltonian with energy relaxation and dephasing. (a)–(f) consider four-qubit Hamiltonian with a finite $({10}^6)$ number of samples. (g) considers eight-qubit Ising spin-chain Hamiltonian with an infinite number of samples. (a),(d) compare the time evolved normalized next-nearest-neighbor correlation function. (b),(e) show the error between the exact value and the practical value. (c),(f) show the fidelity of the effective density matrix ${\rho }_{\mathrm{eff}}^{\alpha }$ and the ideal one ${\rho }_I$ under different error-mitigation scheme $\alpha$. (g) shows that the performance of stochastic and hybrid QEM. Hybrid error-mitigation scheme suppresses the error up to about 4 orders of magnitude even with $10\mathrm{\%}$ model estimation error.

Figure 7

Numerical test of the performance of error-mitigation schemes for the frustrated quantum spin model without model estimation error [(a)–(c)] and with $10\mathrm{\%}$ model estimation error ${\lambda }_{\exp }=1.1{\lambda }_{\mathrm{est}}$ [(d)–(g)]. The parameter settings are the same as in the main text.

Figure 8

Numerical test of the performance of error-mitigation schemes for a two-site quantum system. The system is affected by the time-dependent environmental noise, where the Hamiltonian reads $H=J{\hat{\sigma }}_z^{(1)}{\hat{\sigma }}_z^{(2)}-\sum _{j=1}^2\left[h_j{\hat{\sigma }}_z^{(j)}+{\lambda }^{\mathrm{\prime }}{f}_j(t){\hat{\sigma }}_z^{(j)}\right]$. We consider a low-frequency noise derived in Appendix 20, the dynamical equation is expressed as $\overline{d\rho (t)}/dt\simeq -i[H_0,\rho ]+2{\lambda }^{\mathrm{\prime }2}t\sum _{j=1}^2\left[{\hat{\sigma }}_z^{(j)}\rho {\hat{\sigma }}_z^{(j)}-\rho \right]$. The time-correlated Lindblad noise operator is $L_{\mathrm{dep}}=\sqrt{t}L$, where $L$ has the same form as in Fig. 2. We set the coupling $J=2\pi \times 3$ MHz, field strength $h=0.5J$, noise strength $2{\lambda }^{\mathrm{\prime }2}=0.1$ MHz, time-dependent model estimation error ${\lambda }_{\exp }=1.2{\lambda }_{\mathrm{est}}$, and the sampling numbers ${10}^5$. As proven in Eq. (C8), noise rate is boosted by a factor $r_j^2$, and we use the scaling factor $r_1=1$, $r_2=1.5$ for Richardson extrapolation and the hybrid error mitigation.