Feed-Forward Probabilistic Error Cancellation with Noisy Recovery Gates

We first introduce the notations required to explain the PEC procedure. We consider an $n$ qubits system. Let $\mathcal{Q}$ denote a unitary trace-preserving completely positive (TPCP) map of an ideal quantum circuit on the $n$ qubits system, $\rho_{\text{in}}$ denote the input quantum state to this circuit $\mathcal{Q}$ and $\rho_{\text{out}}$ denote the output quantum state after applying the circuit $\mathcal{Q}$ to the input state $\rho_{\text{in}}$ with a noisy quantum computer. The circuit $\mathcal{Q}$ can be represented as $\mathcal{Q}(\rho)=Q\rho Q^{\dagger}$ with a corresponding unitary matrix $Q$ for any quantum state $\rho$. In addition, the quantum circuit $\mathcal{Q}$ consists of $N_{G}$ ideal quantum gates as follows:

where $\circ$ represents a composition of TPCP maps, and each $\mathcal{U}_{i}$ is a unitary TPCP map corresponding to a quantum gate $U_{i}$. Suppose that a quantum circuit $\mathcal{Q}$ is executed on a noisy quantum computer and that each noise channel $\Lambda_{i}$ is induced for each gate $\mathcal{U}_{i}$, where noise channel $\Lambda_{i}$ can be assumed as a Pauli channel [28, 8]. The output quantum state $\rho_{\text{out}}$ is written as:

We denote an ideal output quantum state as $\rho_{\text{out}}^{\text{ideal}}:=\mathcal{Q}(\rho_{\text{in}})$, then an ideal expectation value of an observable $M$ is written as $\expectationvalue{M}_{\text{ideal}}:=\Tr[M\rho_{\text{out}}^{\text{ideal}}]$.

The objective of PEC is to provide an expectation value with good accuracy while executing a quantum circuit on a noisy quantum computer. The key idea of PEC is to cancel out noises $\Lambda_{i}$ which occurs after $U_{i}$ is executed by adding its inverse $\Lambda_{i}^{-1}$. By cancelling out noises in such a way, PEC can provide an expectation value $\expectationvalue{M}_{\mathcal{Q}}$ via statistical post-processing. To simplify, we describe the PEC process using a single quantum gate $U$. After that, the PEC process is explained for a general quantum circuit. First of all, let $\Lambda$ be a noise induced by operating a quantum gate $\mathcal{U}$. We assume that there exists a gate set $\{\mathcal{R}_{i}\}$ that is implemented on a quantum computer so that the inverse map of noise $\Lambda^{-1}$ can be represented as a linear combination with elements of this set as a basis:

where each $q_{i}$ is a real number (pseudo-probability) and satisfies $\sum_{i}q_{i}=1$ [6]. Then we have

with $\gamma=\sum_{i}\absolutevalue{q_{i}}$ and $\sigma_{i}=\absolutevalue{q_{i}}/\gamma$. Factor $\gamma$ is the sampling overhead, and each $\sigma_{i}$ is the insertion probability of gate $\mathcal{R}_{i}$, and ${\rm sgn}(q_{i})=\pm 1$ means a parity corresponding to gate $\mathcal{R}_{i}$.

Using $\gamma$, the expectation value of observable $M$ can be expressed as follows:

where $\mu_{i}^{\text{eff}}={\rm sgn}(q_{i})m_{i}$ and $m_{i}$ is the measurement outcome of the state $\mathcal{R}_{i}\Lambda\mathcal{U}(\rho_{\text{in}})$, which is generated by applying the gate $\mathcal{R}_{i}$ with probability $\sigma_{i}$ after applying gate $\mathcal{U}_{i}$. Multiplying the outcome $\mu_{i}$ by the parity ${\rm sgn}(q_{i})$ corresponding to $\mathcal{R}_{i}$ gives $\mu_{i}^{\text{eff}}$. The expectation value of the random variable $\gamma\mu^{\text{eff}}$ is approximately an unbiased estimator of $\expectationvalue{M}_{\mathcal{U}}$.

To obtain an expectation value $\expectationvalue{M}_{\mathcal{Q}}$ of an observable $M$ with a quantum circuit $\mathcal{Q}$, we apply the process stated above for $\mathcal{Q}$. Applying the above process to the quantum circuit $\mathcal{Q}=\prod_{k=1}^{N_{G}}\mathcal{U}_{k}$, we have

with $\mathcal{U}_{k}=\gamma^{(k)}\sum_{i_{k}}\mathrm{sgn}(q_{i_{k}})\sigma_{i_{k}}\mathcal{R}_{i_{k}}\Lambda_{k}\mathcal{U}_{k}$. We can obtain a random variable $\mu^{\text{eff}}$ by executing a circuit modified by operation $\mathcal{R}_{i_{k}}$ added with probability $\sigma_{i_{k}}$ and multiplying sign $\prod_{k=1}^{N_{G}}{\rm sgn}(q_{i_{k}})$ to measurement outcome. The expectation value $\expectationvalue{\mu^{\text{eff}}}$ is obtained by repeating this process, and finally we obtain the expectation value $\expectationvalue{M}_{\mathcal{Q}}=\gamma^{\text{tot}}\expectationvalue{\mu^{\text{eff}}}$, where $\gamma^{\text{tot}}=\prod_{k=1}^{N_{G}}\gamma^{(k)}$. The estimator $\expectationvalue{M}_{\mathcal{Q}}$ is unbiased for the exact expectation value $\expectationvalue{M}_{\text{ideal}}$.

Here, operation $\mathcal{R}$ added to cancel noise is called recovery gate in this paper. Note that the recovery gate $\mathcal{R}$ can sometimes introduce noises. To avoid this noise, the recovery $\mathcal{R}$ can be merged with the neighboring gate. However, the gate merging would break the consistency with the PEC process in practical uses. Thus, the gate merging is not performed in PEC.

We introduce depolarizing [24], which is used in our experiments. To explain the noise channel, we denote $\mathcal{I},\mathcal{X},\mathcal{Y}$, and $\mathcal{Z}$ as trace-preserving and completely positive (TPCP) maps corresponding to the Pauli matrices $I,X,Y$, and $Z$, respectively. Similarly, $\mathcal{II},\mathcal{IX},\ldots,\mathcal{ZZ}$ also represent TPCP maps corresponding to couples of Pauli matrices $II,IX,\ldots,ZZ$. In addition, a single qubit state and a two qubits state are denoted as $\rho$ and $\rho^{\prime}$, respectively. The error rate is denoted as $p$.

We explain how to determine an inverse map of depolarizing on a single qubit state for FFPEC. Let $\tilde{\mathcal{D}}_{1}^{-1}$ denote a noisy version of $\mathcal{D}_{1}^{-1}$ as follows:

where $\tilde{\mathcal{X}},\tilde{\mathcal{Y}}$, and $\tilde{\mathcal{Z}}$ are noisy version of $\mathcal{X},\mathcal{Y}$, and $\mathcal{Z}$ for depolarizing, respectively. These are represented as follows:

To obtain the parameter $q$ for FFPEC, equation $\mathcal{\tilde{D}}_{1}^{-1}(\mathcal{D}_{1}(\rho))=\rho$ is solved. Then we have

The sampling cost is $\gamma=(4+p+p^{2})/[(1-p)(4-p)]$ and the insertion probabilities of pauli gates $X$, $Y$, and $Z$ are all the same value $4p/(4+p+p^{2})$.

A modified inverse of depolarizing for two qubits is also determined by a similar way as for a single qubit state explained in the previous section. Let $\tilde{\mathcal{D}}_{2}^{-1}$ denote a noisy version of $\mathcal{D}_{2}^{-1}$ as follows:

where $\widetilde{\mathcal{IX}},\widetilde{\mathcal{IY}},\ldots,\widetilde{\mathcal{ZZ}}$ are noisy versions of Pauli maps corresponding to $\mathcal{IX},\mathcal{IY},\ldots,\mathcal{ZZ}$. Assuming that each $\widetilde{\mathcal{IX}},\widetilde{\mathcal{IY}},\ldots,\widetilde{\mathcal{ZZ}}$ is exposed to two qubits depolarizing noise, they can be represented as follows:

To obtain the parameter $q$, equation $\tilde{\mathcal{D}}_{2}^{-1}(\mathcal{D}_{2}(\rho^{\prime}))=\rho^{\prime}$ is solved. Then we have

The sampling cost is $\gamma=(16+13p+p^{2})/[(1-p)(16-p)]$ and the insertion probabilities of Pauli gates $IX,\ldots$, and $ZZ$ are all the same value $p/(16+13p+p^{2})$. Note that the identity $I$ included in $IX,IY,\ldots,$ and $ZI$ gates is inserted as a recovery gate in this model as well as the standard PEC.

The depolarizing noise model is used with error rates $\{0.001,0.0015,0.002\}$ for single qubit gates and $\{0.01,0.015,0.02\}$ for two qubits gates, which are close to the error rates of current quantum computers. The insertion probabilities of recovery gates for each error rate are summarized in Table I. For each error rate, the insertion probabilities in the FFPEC are slightly larger than those in the PEC, about $0.025\%,0.038\%$ and $0.05\%$ for a single qubit gate, and $0.061\%,0.091\%$ and $0.12\%$ for a two qubits gate. To investigate these effects of the insertion probabilities on the accuracy of expectation values, experiments listed below are addressed using three types of quantum circuits consisting of basic gates as shown in Fig. 2:

• (a)

  investigation of the accuracy of expectation values with error rates $\{0.001,0.0015,0.002\}$ for the $X$ gate with circuit [2],

• (b)

  investigation of the accuracy of expectation values with error rates $\{0.01,0.015,0.02\}$ for the CNOT gate with circuit [2] and,

• (c)

  investigation of the accuracy of expectation values using a circuit consisting of $X$ gates and CNOT gates [2] with error rates $\{(0.001,0.01),(0.0015,0.015),(0.002,0.02)\}$ for the $X$ gate and the CNOT gate.

In the experiments, initial state $\ket{0}^{\otimes 8}$ and the observable $Z^{\otimes 8}$ are used. The exact expectation values of the observable for each circuit are all $1$ to verify the influence of noise for expectation values obtained using the PEC and FFPEC. $10^{3}$ of expectation values are sampled with $10^{6}$ shots to evaluate the accuracies (averages and standard deviations) of sampled expectation values using the PEC and FFPEC.

To more precisely evaluate the performance of the PEC and FFPEC, their analytical expectation values are used for each circuit in Fig. 2. The analytical expectation values are obtained using the following TCPC map, which is included the noise $\Lambda$ to the recovery gate $\mathcal{R}$ based on Eq. (6):

Table II shows the sampling overhead of the PEC and FFPEC for each error rate with respect to single qubit gates, two qubits gates, and circuits 2, 2, and 2. The sampling overhead corresponds to the standard deviation, where both PEC and FFPEC have almost the same standard deviation.

Figures 3, 4, and 5 show mitigation results for circuits 2, 2, and 2, respectively. The horizontal axis represents error rates, and the vertical axis represents expectation values obtained using the PEC and FFPEC. The red lines show the numerical results of PEC and the green lines show those of FFPEC by samplings. The blue line represents analytical expectation values obtained by PEC, while the black dashed line represents those of FFPEC. Although the analytical expectation values of PEC do not match the exact expectation value of $1$ due to the influence of noise by inserting the recovery gate, the analytical expectation value of FFPEC matches the exact one. Here, the scale on the vertical axis of all results in three figures is consistent to allow comparison of analytical expectation values with those of numerical ones. In Fig. 3, the standard deviations are $0.03680$ at error rate $0.0015$ and $0.1231$ at error rate $0.002$. In all results, the standard deviations of the PEC and FFPEC are almost the same because they have the same sampling overhead summarized in Table II. In addition, both the PEC and FFPEC can provide improved expectation values as shown in Figs. 3, 4, and 5, where expectation values without mitigation are summarized in Table III.

The results from the PEC and FFPEC shown in Fig. 3 show that, as the error rate increases, the standard deviations increase and the averages shift away from the exact expectation value 1. This is because $10^{6}$ shots are not enough for convergence of expectation values of the circuit 2 with large sampling overheads. For the error rate $0.001$, which is approximately that of current quantum computers of single qubit gate, both the PEC and FFPEC have the almost same absolute error from the exact value of $1$, which is $9.256\times 10^{-4}$. This observation implies that the performance of FFPEC is the same as that of the PEC. If more than $10^{6}$ shots are used, FFPEC can provide a more accurate expectation value than the PEC.

In the results shown in Fig. 4, the average values obtained using both the PEC and FFPEC converge to their analytical expectation values, respectively. Focusing on the error rate $0.01$, which is approximately that of current quantum computers of two qubits gate (e.g. CNOT gate), the absolute error between the exact expectation value and the average one is $3.858\times 10^{-4}$ for the PEC and $5.847\times 10^{-5}$ for FFPEC. Thus, FFPEC can provide a more accurate expectation value than the PEC when circuits like Fig. 2 are used.

In the results shown in Fig. 5, the average values obtained using PEC converge to their analytical expectation values. The average values of FFPEC also converge to the analytical expectation values, except for the result at the error rate $(0.002,0.02)$. The reason why the result at the error rate does not converge to the analytical expectation value is the shortage of shots. Focusing on the error rate $(0.001,0.01)$, which is approximately that of current quantum computers of single and two qubits gate, the absolute error between the exact expectation value and the average value is $2.743\times 10^{-4}$ for the PEC and $1.084\times 10^{-4}$ for FFPEC. Thus, FFPEC can provide a slightly more accurate expectation value than the PEC.

Since practical quantum circuits consist of single qubit gates and CNOT gates, as shown in the circuit 2, the results of the error rate $(0.001,0.01)$ in Fig. 5 are expected results on current quantum computers, although thermal relaxation errors and readout errors exist. FFPEC can provide slightly more accurate expectation values than the PEC as stated above. The error mitigation capability of the PEC is limited due to the noise induced by inserting recovery gates, but FFPEC has no such limitation. Thus, the more shots are taken to obtain a highly accurate expectation value, FFPEC can provide a more accurate one than the PEC. For the practical implementation of FFPEC, there are challenges: (i) to address other errors such as thermal relaxation errors and readout errors, (ii) to improve error inverse maps based on the working of gates on real hardware such as the identity gate is not executed, a two qubits gate is executed as per single qubit gate if the two qubits gate can be decomposed to its tensor product, the $Z$ gate can be performed as virtual [31], and (iii) to reduce the number of shots to provide expectation values with the required accuracy. In addition, both the PEC and FFPEC can sometimes yield expectation values that exceed the range $[-1,1]$, as shown in the numerical experiments For algorithms utilizing the PEC and FFPEC, it would be unsuitable to use values outside this range. Accordingly, it would be better to correct to $1$ or $-1$ when such values are obtained.