Learning-Based Quantum Error Mitigation

If noisy-intermediate-scale-quantum-era quantum computers are to perform useful tasks, they will need to employ powerful error mitigation techniques. Quasiprobability methods can permit perfect error compensation at the cost of additional circuit executions, provided that the nature of the error model is fully understood and sufficiently local both spatially and temporally. Unfortunately, these conditions are challenging to satisfy. Here we present a method by which the proper compensation strategy can instead be learned ab initio. Our training process uses multiple variants of the primary circuit where all non-Clifford gates are substituted with gates that are efficient to simulate classically. The process yields a configuration that is near optimal versus noise in the real system with its non-Clifford gate set. Having presented a range of learning strategies, we demonstrate the power of the technique both with real quantum hardware (IBM devices) and exactly emulated imperfect quantum computers. The systems suffer a range of noise severities and types, including spatially and temporally correlated variants. In all cases the protocol successfully adapts to the noise and mitigates it to a high degree.

Popular Summary

Quantum computers are “noisy”: errors sneak in whenever more than a few qubits perform a calculation of any significant complexity. For near-term devices, we can try to mitigate errors, minimizing their impact so that the output is still useful. However, there is a problem. The most powerful error mitigation processes require the user to know the exact nature of the noise that the computer suffers from and that knowledge is hard to get. The task of learning all about the noise in the system can itself become an insurmountable problem. We want to sidestep this requirement.

When machine learning is used to, for example, learn to recognize handwritten postal codes, it does so without ever having to actually describe the countless variations of human handwriting. In the present paper we find that we can also automatically learn the key features of quantum noise provided we have good training examples. We take the circuit we would like to run on the quantum device and create many training variants with the special property that they can be evaluated on a conventional computer, so that we know what the “right” output is.

We prove that the lessons learned on the simplified circuits will work on the real one. To check, we use high-performance conventional computers to “pretend” to be quantum computers with various types of noise problem, and then we successfully apply our learning technique. Our approach can provide a new pathway toward the milestone of “quantum advantage,” the moment when quantum machines will do something really useful.

Figure 1

Simple example of a circuit without and with error mitigation. Each circuit has four qubits and two layers of frame gates $G_1$ and $G_2$ (dashed boxes). Note that errors afflict the frame gates and may correlate over arbitrarily many qubits within a box and between boxes (i.e., spatial and temporal errors). Frame operations (orange) include the qubit initialization, frame gates, and measurement. There are three layers of computing gates $R_i$ (blue) in each circuit. To implement the error mitigation, two layers of Pauli gates $P_i$ (brown) are introduced before and after each layer of computing gates. Usually, single-qubit gates next to each other in the circuit can be combined into one single-qubit gate in the physical implementation.

Figure 2

Empirical cumulative distribution function of estimated $\mathrm{\Delta }⟨Z_1⟩$ for 500 pseudorandom circuits with spatially correlated dephasing noise (a) and spatially correlated depolarizing noise (b). Results for circuits without error mitigation (black), with tomographic error mitigation (TEM; red), and with learning-based error mitigation (LBEM; green) are presented. Additionally, we include the results for learning-based error mitigation when the sample size $M\to \mathrm{\infty }$ (dashed green).

Figure 3

Empirical cumulative distribution function of estimated $\mathrm{\Delta }⟨Z_1⟩$ for 500 pseudorandom circuits with temporally correlated dephasing noise. Results for circuits without error mitigation (black), with tomographic error mitigation (red), and with learning-based error mitigation (green) are presented. Additionally, we include the results for learning-based error mitigation when the sample size $M\to \mathrm{\infty }$ (dashed green).

Figure 4

Error model after each set of single-qubit gates (blue) and two-qubit gates (black). Unitary single-qubit gates $U$ may be either single-qubit Clifford gates or arbitrary rotations around the $y$ axis of the Bloch sphere. Here $\gamma$ (orange) describes the amplitude damping channel, while $\mathcal{D}$ (red) describes a biased dephasing and depolarizing channel [the channel is described in the main text, Eq. (9)].

Figure 5

Expectation values of $⟨Z_1⟩$ obtained from a single experiment repeated 10 000 times with no error mitigation (blue), tomography-based error mitigation (green), and learning-based error mitigation (orange). Dashed line indicates an error-free expectation value $⟨Z_1{⟩}^{\mathrm{EF}}$. The solid lines describe analytically derived probability distributions for each approach.

Figure 6

Performance of the learning-based mitigation protocol in the context of a QVA. The ansatz circuit (a) includes $20$ parameterized gates $R_y({\theta }_i)=\exp (-i{\theta }_i/2{\sigma }_y)$ and the parameters are adjusted with the goal of finding the ground state of a certain frustrated spin system. The processor is a virtual noisy four-qubit device, emulated by the QuEST. When the QVA employs simple extrapolation-based mitigation [(c), orange line], the final energy is above the ideal target by $5.6\mathrm{\%}$. Instead using learning-based error mitigation, the final energy is below the target by only $0.7\mathrm{\%}$. The parameter evolution for the latter case is shown in panel (b). Further details of the two protocols are provided in the main text.

Figure 7

Empirical cumulative distribution function of estimated $\mathrm{\Delta }⟨Z_1⟩=|⟨Z_1⟩-⟨Z_1{⟩}^{\mathrm{EF}}|$ for $500$ configurations of randomly generated computing gates $\mathbf{R}$. Each computing gate is uniformly sampled from the single-qubit unitary group according to the Haar measure. For each computing-gate configuration, $M=10000$ random configurations of error-correcting gates $\mathbf{P}$ are generated to evaluate the error-mitigated result. In the inset in (d), we show $⟨Z_1⟩$ of $100$ configurations for the eight-qubit circuit. Error bars represent estimated standard errors.

Figure 8

Average error rescaling factor for various circuit sizes. The circuit size is equal to the number of qubits and layers in the circuit. For each circuit layout, we randomly generate $1000$ configurations of Clifford computing gates $\mathbf{R}$. We choose the configuration such that the error-free computation result is nonzero. For each computing-gate configuration, $M=1000000$ random configurations of error-correcting gates $\mathbf{P}$ are generated to evaluate the error-mitigated result.

Figure 9

Two-qubit DQCp circuit used in the experimental demonstration of the learning-based quantum error mitigation protocol.

Figure 10

Computation results of the two-qubit DQCp circuit, obtained from three different quantum hardware. Here $⟨Z⟩=\mathrm{com}(e^{-i\theta Z/2},I)$ is the raw result without error mitigation and $⟨Z⟩={\mathrm{com}}^{\mathrm{EM}}(e^{-i\theta Z/2},I)$ is the error-mitigated result.

Figure 11

Ground-state energy surface of $H_2$ in the minimal basis computed using the variational quantum eigensolver. The blue solid line is computed using package Qiskit. Square and triangular scatters represent results without and with the learning-based quantum error mitigation computed on ibmq_santiago.

Figure 12

Circuit used in the error-mitigated variational quantum eigensolver. The gates $R_x(\varphi )=e^{-i\varphi X/2}$.

Figure 13

The computing without and with errors. From left to right, $\mathcal{R}={\mathcal{R}}_1,{\mathcal{R}}_2,\dots ,{\mathcal{R}}_{N+1}$, $\mathcal{P}={\mathcal{P}}_1,{\mathcal{P}}_2,\dots ,{\mathcal{P}}_{2N+2}$, ${\mathcal{G}}^{\mathrm{EF}}={\mathcal{G}}_1^{\mathrm{EF}},{\mathcal{G}}_2^{\mathrm{EF}},\dots ,{\mathcal{G}}_N^{\mathrm{EF}}$, and $\mathcal{G}={\mathcal{G}}_1,{\mathcal{G}}_2,\dots ,{\mathcal{G}}_N$. Here $\mathcal{I}=[{\mathbb{1}}_E]$ is the identity map on the environment; $S$ denotes system and $E$ denotes environment.

Figure 14

The error-free frame-operation tensor ${\mathcal{F}}^{\mathrm{EF}}$ and the erroneous frame-operation tensor $\mathcal{F}$. The arrows denote the direction of the time. Along the direction of arrows, $\mathcal{R}={\mathcal{R}}_1,{\mathcal{R}}_2,\dots ,{\mathcal{R}}_{N+1}$, $\mathcal{P}={\mathcal{P}}_1,{\mathcal{P}}_2,\dots ,{\mathcal{P}}_{2N+2}$, ${\mathcal{G}}^{\mathrm{EF}}={\mathcal{G}}_1^{\mathrm{EF}},{\mathcal{G}}_2^{\mathrm{EF}},\dots ,{\mathcal{G}}_N^{\mathrm{EF}}$, and $\mathcal{G}={\mathcal{G}}_1,{\mathcal{G}}_2,\dots ,{\mathcal{G}}_N$. Here $S$ denotes system and $E$ denotes environment.

Figure 15

Eight-qubit wide and eight-layer deep circuit layout. Gates $U$ represent single-qubit unitary gates, and the two-qubit gates are controlled-not gates. For circuits in $\mathbb{T}$, gates $U$ are all Clifford. The initial state is $|0{⟩}^{\otimes n}$, and the measurement is done in the $Z$ basis on the bottom qubit.

Figure 16

Normalized inner product $q_x.q_{20}/(|q_x||q_{20}|)$ between quasiprobability distributions that are obtained in the learning part of the protocol, as described in Sec. 7, with Clifford overhead constants $c=x$ ($q_x$) and $c=20$ ($q_{20}$). We plot the inner product for different circuit sizes $nxn$, where $n$ is both the number of qubits and the number of layers of two-qubit gates. Here we have chosen $q_{20}$ as the reference for other distributions, assuming that $q_{20}$ deviates only marginally from $q_{\mathrm{opt}}^{\prime }$, which is obtained from the full training set $\mathbb{T}$. This allows us to estimate the impact on errors due to the truncation of the training set and, hence, lets us choose a suitable Clifford overhead constant $c$ for our simulations.

Figure 17

Empirical cumulative distribution function of estimated $\mathrm{\Delta }⟨Z_1⟩$ for 500 pseudorandom circuits with spatially correlated dephasing noise. Results for circuits without error mitigation (black), with tomographic error mitigation for $k=1$ (dashed red) and $k=2$ (red), with a single-parameter learning-based error mitigation (orange), and with a multiparameter learning-based error mitigation (green) are presented.

Figure 18

Qubits follow the cycle graph pattern with the first qubit being adjacent to the last one. Gates are implemented from bottom up if they are on the same vertical line. Black two-qubit gates are control-$Z$ gates and single qubit $R_y$ gates are rotations around the $y$ axis of the Bloch sphere. Here $M_z$ denotes a single-qubit measurement along the z axis of the Bloch sphere.

Figure 19

Values of the loss function in the gradient descent. We use $n\times N$ to denote an $n$-qubit, $N$-layer circuit. For each circuit size, we randomly generate $N=1000$ computing-gate configurations $\mathbf{R}$, and for each computing-gate configuration, we randomly generate $M=1000$ configurations of error-correcting gates $\mathbf{P}$. Note that the estimated cost function does not converge to zero with a finite number of samples due to a biased estimator.