Abstract
Various quantum applications can be reduced to estimating expectation values, which are inevitably deviated by operational and environmental errors. Although errors can be tackled by quantum error correction, the overheads are far from being affordable for nearterm technologies. To alleviate the detrimental effects of errors on the estimation of expectation values, quantum error mitigation techniques have been proposed, which require no additional qubit resources. Here we benchmark the performance of a quantum error mitigation technique based on probabilistic error cancellation in a trappedion system. Our results clearly show that effective gate fidelities exceed physical fidelities, i.e., we surpass the breakeven point of eliminating gate errors, by programming quantum circuits. The error rates are effectively reduced from (1.10 ± 0.12) × 10^{−3} to (1.44 ± 5.28) × 10^{−5} and from (0.99 ± 0.06) × 10^{−2} to (0.96 ± 0.10) × 10^{−3} for single and twoqubit gates, respectively. Our demonstration opens up the possibility of implementing highfidelity computations on a nearterm noisy quantum device.
Introduction
Quantum computers^{1} can extend classical computational reach in diverse research fields, including quantum chemistry, material science, and even machine learning. Based on various technological advances so far, such nontrival quantum applications have been pursued with currently available devices mainly through quantumclassical hybrid schemes^{2,3}. The schemes combine the advantages of classical and quantum computation, where quantum processors are used to estimate expectation values of physical observables on certain states for classical feedback. The hybrid schemes can be applied to estimate the groundstate energies of molecules^{3,4,5}, to simulate quantum models in materials^{6} and highenergy physics^{7}, and to find approximate solutions of optimization problems^{8}. Although it is anticipated that around a hundred wellbehaved qubits are required for such schemes to outperform current classical counterparts in quantum chemistry^{9,10,11}, the advantages are only possible with accurate quantum processors. However, expectation values obtained with output results of the quantum devices are inevitably deviated because of errors originated from both environmental fluctuations and operational imperfections. Therefore, techniques for accurately estimating expectation values with improving the accuracy of noisy quantum processors are of great importance.
Apart from physically improving the devices, the deviations in estimating expectation values can be suppressed on the algorithmic level. For example, quantum error correction^{12,13} provides a mean of faulttolerant quantum computation, which results in accurate expectation values. However, quantum error correcting codes require complex coding schemes, a large number of physical qubits, and low error rates, which are still far from being affordable for nearterm quantum technologies^{14,15}. Consequently, it has not yet been demonstrated that quantum fault tolerance protocols can increase the fidelity of computation operations in any physical implementation. Alternatively, for the quantum algorithms estimating expectation values, the reliability of computation result can be improved by recently proposed error mitigation schemes^{16,17,18,19,20} without challenging requirements for quantum error corrections. The probabilistic errorcancellation method provides a comprehensive way to mitigate errors in expectation estimation tasks^{17,18,21}. It begins with characterizing imperfect operations on the quantum device by tomography technique and then cancels errors by sampling random quantum circuits, according to a quasiprobability distribution derived from reconstructing ideal quantum operations with characterized imperfect ones. Please note that this method does not improve the physical quality of quantum states or gates but reduces the error in the estimation of expectation values.
Here we construct a trappedion system with full controllability and investigate the universal validity of the probabilistic errorcancellation method in a general quantum computational context. We apply the method to every imperfect elementary quantum operation and benchmark the performance of errormitigated quantum computation^{22}. We observe singnificant improvements on effective gate fidelities of single and twoqubit gates by an order of magnitude to those of physical gates. Here, the effective gate fidelities are obtained by fitting the corresponding expectation values estimated with error mitigation, which are not actual physical gate fidelities.
Results
Paradigm of errormitigated quantum computation
The paradigm of errormitigated quantum computation is shown in Fig. 1. The noisy quantum device is treated as a multiqubit black box in Fig. 1a, capable of preparing each qubit into an initial state ρ_{0}, performing a set of singlequbit and twoqubit gates, and twooutcome measurement on each qubit, which is described by a positive operatorvalued measure \({\mathcal{M}}\equiv \{{E}_{0},I{E}_{0}\}\) with I being the 2 × 2 identity operator. These quantum operations are generally not accurate because of errors from operational imperfections and environmental fluctuations. As proposed in ref. ^{18}, we perform the gate set tomography^{23,24,25} and characterize state preparation and measurement (SPAM) and gates of noisy quantum devices by Gram matrices and Pauli transfer matrices (PTMs), respectively^{25}, as shown in Fig. 1b. When we repeatedly execute a quantum circuit with such a noisy device aiming at obtaining the expectation values of observables of interest, the estimation will be deviated from the ideal case due to the imperfection of the quantum device, as shown in Fig. 1c. The correction of each noisy quantum operation can be decomposed to the combination of experimental basis operations (which we give later) with quasiprobabilities as shown in Fig. 1d. As some of the quasiprobabilities can be negative, we cannot physically implement the decomposition. However, the expectation of the decomposition can be estimated by sampling circuits with random basis operations according to the quasiprobabilities^{17,18}. After running the random circuits with the corrections, the probability distribution of the output expectation value is shifted towards the ideal value at a cost of enlarged variance due to the presence of negative values in the quasiprobabilities^{18}, as shown in Fig. 1c. The variance can be reduced by increasing the repetition number, which is the number of randomcircuit instances.
Experimental realization
In our experimental realization, the quantum hardware encapsulated in the black box is a trappedion system, where ^{171}Yb^{+} ions are trapped into a linear crystal and individually manipulated by global and individual laser beams, as shown in Fig. 1a. To encode quantum information, a pair of clock states in the groundstate manifold ^{2}S_{1∕2}, i.e., \(\leftF=0,{m}_{F}=0\right\rangle\) and \(\leftF=1,{m}_{F}=0\right\rangle\), are denoted as the computational basis \(\left\{\left0\right\rangle ,\left1\right\rangle \right\}\) of a qubit. At the beginning of executing a quantum circuit, each ion qubit is initialized to \(\left0\right\rangle\) by optical pumping. We implement singlequbit operations by Raman laser beams with beatnote frequency about the hyperfine splitting ω_{0} = 2π × 12.642821 GHz. In addition, the twoqubit operation, i.e., the MølmerSørensen YYgate (MS_{YY}) is realized by driving transverse motional modes^{26,27}, with frequencies in the xdirection {ν_{1}, ν_{2}} = {1.954, 2.048} MHz. We apply amplitudeshaped^{28} bichromatic Raman beams with beatnote frequencies ω_{0} ± μ, where μ is set to be the middle frequency of the two motional modes, and achieve the MS_{YY} gate for 25 μs. We also realize the MS ZZgate (MS_{ZZ}) by adding singlequbit rotations before and after the MS_{YY} gate^{29} (see Supplementary Fig. 4b). At the end of the execution, internal states of qubits are measured by statedependent fluorescence detection^{30}. It is noteworthy that to collect fluorescence photons, we use a photomultiplier tube in the singlequbit case and an electronmultiplying chargecoupled device (EMCCD) in the twoqubit case.
Characterization of quantum device
We introduce the PTM representation for the mathematical description of an nqubit noisy quantum device, where density operators ρ and physical observable E are represented by 2^{n}entry column vectors \(\left\left.\rho \right\rangle \right\rangle\) and row vectors \(\left\langle \left\langle E\right.\right\), and quantum gates G are represented by 2^{2n} × 2^{2n} PTMs R_{G}. Here, the expectation value of the observable \(\hat{E}\) after operating G_{s} on the initial state \(\hat{\rho }\) is represented by 〈〈E∣R_{G}∣ρ〉〉. PTMs can be determined by gate set tomography, which requires informationally complete data obtained from experiments with initial states from a basis set \({{\mathcal{S}}}_{n}\equiv {\{\left0\right\rangle ,\left1\right\rangle ,{\left1\right\rangle }_{X},{\left1\right\rangle }_{Y}\}}^{\otimes n}\) and measurement of the observables from the nqubit Pauli basis \({{\mathcal{P}}}_{n}={\{I,X,Y,Z\}}^{\otimes n}\), where \({\left1\right\rangle }_{X}\) and \({\left1\right\rangle }_{Y}\) are the eigenstates of Puali operators X and Y, respectively. Compared with quantum process tomography, gate set tomography is featured by appropriately taking consideration of SPAM errors, which is of great importance in quantum computations with high accuracy. In gate set tomography, the states in \({{\mathcal{S}}}_{n}\) and the measurement of observables in \({{\mathcal{P}}}_{n}\) are realized by using a set of fiducial gates \({{\mathcal{F}}}_{n}\equiv {\{I,{X}_{\pi },{Y}_{\frac{\pi }{2}},{X}_{\frac{\pi }{2}}\}}^{\otimes n}\) consisting of the identity operation and the X or Y axis rotations on each qubit, which are to be characterized together with the rest of the quantum operations. The singlequbit SPAM errors are reflected in the Gram matrix^{25}, as shown in Fig. 2a, which is obtained by preparing the qubit in one of the states \({{\mathcal{S}}}_{1}\), \(\left\left.{\rho }_{i}\right\rangle \right\rangle ={R}_{{F}_{i}}\left\left.{\rho }_{0}\right\rangle \right\rangle\), and measuring the expectation values of the operators in the singlequbit Pauli basis \({{\mathcal{P}}}_{1}\), \(\left\langle \left\langle {E}_{i}\right.\right=\left\langle \left\langle {E}_{0}\right.\right{R}_{{F}_{i}}\), where ρ_{0} and E_{0} are ideally associated with \(\left0\right\rangle \left\langle 0\right\) and Z, respectively.
For singlequbit randomized benchmarking^{22}, we design pulse sequences for implementing majoraxis π pulses {X_{±π}, Y_{±π}, Z_{±π}} and \(\frac{\pi }{2}\) pulses \(\{{X}_{\pm \! \frac{\pi }{2}},{Y}_{\pm \! \frac{\pi }{2}}\}\). Thus, the gate set for the singlequbit case is \({{\mathcal{G}}}_{1}=\{I,{X}_{\! \pm \! \pi },{Y}\! _{\pm \! \pi },{Z}\! _{\pm \! \pi },{X}\! _{\pm \! \frac{\pi }{2}},{Y}\! _{\pm \! \frac{\pi }{2}}\}\), where I is the identity operation. The gate set for implementing twoqubit random circuits are \({{\mathcal{G}}}_{2}={{\mathcal{G}}}_{1}^{\otimes 2}\cup \{{{\rm{MS}}}_{YY},{{\rm{MS}}}_{ZZ}\}\). We experimentally obtain the PTMs of all the gates in the gate set by maximizing a likelihood function with the assumption that Pauli errors are dominant in our devices (see Methods).
The reconstructed PTMs of \({X} \!_{\pm \! \frac{\pi }{2}}\) and \({Y}\!_{\pm \! \frac{\pi }{2}}\) for the singlequbit case and those of MS_{YY} and MS_{ZZ} gates for the twoqubit case are shown in Fig. 2b, c, respectively (more data for the singlequbit case are in Supplementary Fig. 1a). We note that, for the gate set tomography of two qubits, we apply a twostep parameter estimation, as the infidelities for the singlequbit gates are about an order lower than those of the twoqubit gates. We first determine the Pauli error rates for all the singlequbit gates in \({{\mathcal{G}}}_{1}^{\otimes 2}\) as described above and then characterize the twoqubit gate MS_{YY} based on the knowledge of the characterized singlequbit gates (see Methods). The MS_{ZZ} gate is derived from those results. Using these reconstructed PTMs, we numerically simulate the singlequbit randomized benchmarking and twoqubit random circuits on a classical computer. The comparisons between the numerically reconstructed and experimental data clearly validate the Pauli error assumption within both error bars (see Supplementary Fig. 2).
The initial state, quantum gates, and measurement are deviated from the ideal ones, as experimentally characterized by Gram matrix and PTMs. Mathematically, we can reconstruct the ideal ones by a weighted combination of experimental operations^{17,18}. As we cannot distinguish errors in state preparation from those in measurement, we ascribe all of the SPAM errors to state preparation and decompose the initial state \(\left\left.{\rho }_{0}^{{\rm{id}}}\right\rangle \right\rangle ={\sum }_{i}{q}_{0,i}\left\left.{\rho }_{i}\right\rangle \right\rangle\). The quasiprobabilities q_{0,i} for the decomposition of the ideal singlequbit initial state is shown in Fig. 3a. It is noteworthy that for the twoqubit case, the SPAM errors are much more serious because of the EMCCD and we calibrate the results to remove the SPAM errors as proposed in ref. ^{31}. We prepare the system in the states \(\left00\right\rangle\) and \(\left11\right\rangle\), and measure the state fidelities of \(\left0\right\rangle\) and \(\left1\right\rangle\) for both qubits. The infidelities of these states give the SPAM error probability associated with each measurement outcome, which can then be used to remove the SPAM errors by data processing.
An ideal quantum gate \({G}_{\mathrm{{s}}}^{{\rm{id}}}\) can be written as the experimental one followed by the inverse of noise operation, i.e., \({{R}_{{G}_{\mathrm{{s}}}^{{\rm{id}}}}={N}_{\mathrm{{s}}}^{1}{R}_{{G}_{\mathrm{{s}}}}}\), where the noise operation N_{s} introduces errors in the experimental gate \({R}_{{G}_{\mathrm{{s}}}}={N}_{\mathrm{{s}}}{R}_{{G}_{\mathrm{{s}}}}^{{\rm{id}}}\). The inverse of the noise operation \({N}_{\mathrm{{s}}}^{1}\) is then decomposed by the experimental operations associated with the nqubit Pauli group, \({N}_{\mathrm{{s}}}^{1}={\sum }_{j}{q}_{s,j}{R}_{{P}_{j}}\) with Pauli error assumption, where the quasiprobabilities q_{s,j} are determined by a set of linear equations. We show decompositions of the inverse error operations for singlequbit gates \(\{{X}\! _{\pm \! \frac{\pi }{2}},{Y}\!\! _{\pm \! \frac{\pi }{2}}\}\) in Fig. 3b (more data in Supplementary Fig. 1b) and for twoqubit gates {MS_{YY} and MS_{ZZ}} in Fig. 3c.
Benchmarking of the quantum error mitigation protocol
We benchmark the performance of the quantum error mitigation using a set of random computations, in the spirit of randomized benchmarking. Each specific computation starts with fully polarized initial states, \(\left0\right\rangle\) in the singlequbit case and \(\left00\right\rangle\) in the twoqubit case, and ends with measuring Z on each qubit. Between the SPAM, there is a sequence of randomly selected quantum gates. We note that the randomness in selecting the gate sequence is for the purpose of benchmarking the performance rather than correcting errors. For each specific computation, i.e., gate sequence, we apply the error mitigation and modify the gate sequence with random basis operations to correct errors. We remark that, for each specific computation, we observe the improvement on the computation accuracy by using the error mitigation.
For the singlequbit case, benchmarking computations are selected according to the standard randomized benchmarking, i.e., a gate sequence of length L contains L computational gates and L + 1 interleaving identity or Pauli operations, uniformly drawn from the set \(\{{X}\! _{\pm \! \frac{\pi }{2}},{Y}\! _{\pm \! \frac{\pi }{2}}\}\) and {I, X_{±π}, Y_{±π}, Z_{±π}}, respectively. For each sequence length L, we choose four sequences whose ideal final states are the eigenstates of the Pauli Z operator. We then repeatedly implement each of the sequences with a trappedion system consisting of a single trapped ion and measure the state fidelity between the ideal and experimentally prepared final states. In Fig. 4a, we show the dependence of the average fidelity without error mitigation, obtained by averaging the state fidelities over sequeces of the same length, on the sequence length. We numerically fit the average fidelity with an exponential function and obtain the error rate per singlequbit gate as (1.10 ± 0.12) × 10^{−3}.
To obtain unbiased estimator of the expectation value, both the initial state and 2L + 1 gates in the selected sequence need to be decomposed and resampled, where the initial state is replaced probabilistically by one of the states in \({{\mathcal{S}}}_{1}\), and each experimental gate is followed by a random Pauli or identity operation drawn from \({{\mathcal{P}}}_{1}\). Thus, for a specific computation with (2L + 1) gates, there are 4^{2L+2} possible experimental settings. As the number of settings grows exponentially with the length of the random sequence, we use the MonteCarlo method to compute the result by sampling random experimental settings, which are specified by an index i for the initial state \(\left\left.{\rho }_{i}\right\rangle \right\rangle\) and two (2L + 1)entry index vectors \({\bf{a}}\equiv \left({a}_{1},\ldots ,{a}_{2L+1}\right)\) and \({\bf{b}}\equiv \left({b}_{1},\ldots ,{b}_{2L+1}\right)\) specifying the computation and the choices of the errorcompensating operations. We note that for a specific computation, a is determined but b is random. The probability of an experimental setting \(\langle \langle {E}_{0} {\prod }_{l=1}^{2L+1}{R}_{{P}_{{b}_{l}}}{R}_{{G}_{{a}_{l}}} {\rho }_{i}\rangle \rangle\), where \({G}_{{a}_{l}}\in {{\mathcal{G}}}_{1}\) and \({P}_{{b}_{l}}\in {{\mathcal{P}}}_{1}\), is \({C}^{1}\left{q}_{0,i}\left({\prod }_{l=1}^{2L+1}{q}_{{a}_{l},{b}_{l}}\right)\right\). Here, the rescaling factor \(C={\sum }_{i,\ldots ,\left({a}_{l},{b}_{l}\right),\ldots }\left{q}_{0,i}\left({\prod }_{l=1}^{2L+1}{q}_{{a}_{l},{b}_{l}}\right)\right\ge 1\) characterizes the cost to mitigate the errors. It is noteworthy that the signs of the coefficients, i.e., \({\rm{sgn}}\left[{q}_{0,i}\left({\prod }_{l=1}^{2L+1}{q}_{{a}_{l},{b}_{l}}\right)\right]\), are integrated into the measurement results of the random experiments (see Methods). In Fig. 4a, we represent the errormitigated singlequbit randomized benchmarking with length L up to 64 and show that the singlequbit gate error rate is effectively suppressed to (1.44 ± 5.28) × 10^{−5}.
For the twoqubit case, we select four gate sequences as benchmarking computations for each length L. Each sequence contains L twoqubit gates uniformly drawn from the set {MS_{YY}, MS_{ZZ}} with interleaving singlequbit gates^{32}. The sequence is selected under the restriction that the ideal final state is an eigenstate of Z^{⊗2}. Similar to the singlequbit case described above, we apply error mitigation to each of the twoqubit gate sequences with length L up to 6 and represent the errormitigated results in Fig. 4b, where the twoqubit gate error rate is effectively suppressed from (0.99 ± 0.06) × 10^{−2} to (0.96 ± 0.10) × 10^{−3}.
Discussion
Our work shows that for the estimation of expection values, the error mitigation technique, i.e., probabilistic error cancellation^{17,18,21}, surely have the capacity of surpassing the breakeven point, where the effective gates are superior to their physical building blocks, at an affordable cost with respect to nearfuture quantum techniques. We note that error mitigation techniques are developed for the intermediatescale quantum computation. The cost of the error mitigation increases with the circuit depth; therefore, techniques such as quantum error correction are still needed for largescale faulttolerant quantum computation. The effective infidelity after error mitigation comes from the Pauli error assumption, timedependent systematic drifting^{33} for both singlequbit and twoqubit cases, and crosstalk error of singlequbit addressing operations for the twoqubit case (see Methods). Thus, further improvement requires both calibrating and stabilizing the quantum device. With technologies to tackle the crosstalk error, the probabilistic errorcancellation method of quantum error mitigation can be straightforwardly applied to systems with more qubits for realizing highfidelity quantum computation.
Methods
Maximumlikelihood gate set tomography
To obtain the PTMs of all the gates in the gate set, we experimentlly measure informationally complete data consisting of the average \({\bar{m}}_{ijk}\) and variance Δ_{ijk} of the expectation value \(\langle \langle {E}_{i} {R}_{{G}_{j}} {\rho }_{k}\rangle \rangle\), which are obtained by repeating the corresponding experimental settings enough number of times. We assume Pauli errors are dominant in our device, where each of the noisy quantum gate \({G}_{j}\in {{\mathcal{G}}}_{n}\) is modeled with the ideal gate \({G}_{j}^{{\rm{id}}}\) followed by a Pauli error channel. We use a maximumlikelihood estimation for the reconstruction of PTMs of all the gates in the gate set, parameterized as ansatz \({R}_{{G}_{j}}={N}_{j}{R}_{{G}_{j}}^{{\rm{id}}}\), where \({N}_{j}={\sum }_{l}{p}_{j,l}{R}_{{P}_{l}}\), with variational parameters being gatespecific Pauli error rates p_{j,l}. With the ansatz for each gate, we calculate the ansatz prediction for the expectation value of each experimental setting, denoted as m_{ijk}. We then define the following likelihood function^{25},
which takes its maximum value when the experimental average values \({\bar{m}}_{ijk}\) and the ansatz expectations m_{ijk} coincide with each other. Thus, the gatespecific Pauli error rates can be determined by maximizing the likelihood function, with which we construct the PTMs of the imperfect gates that are implementable in the quantum device.
Characterization and decomposition of singlequbit gate set
We use gate set tomography to characterize the singlequbit operations. In the superoperator formalism, each experimental singlequbit operation \({R}_{{G}_{\mathrm{{s}}}}\) can be describe as an ideal 4 × 4 PTM followed by a PTM of noise operation N_{s}. With Pauli error assumption, each N_{s} can be written as \({N}_{\mathrm{{s}}}={p}_{s,0}{R}_{I}^{{\rm{id}}}+{p}_{s,1}{R}_{X}^{{\rm{id}}}+{p}_{s,2}{R}_{Y}^{{\rm{id}}}+{p}_{s,3}{R}_{Z}^{{\rm{id}}}\), where p_{s,j} are the Pauli error rates and ∑_{j}p_{s,j} = 1 for tracepreserving condition. As there are 11 gate in \({{\mathcal{G}}}_{1}\), \({{\mathcal{F}}}_{1}\subset {{\mathcal{G}}}_{1}\) and the experimental initial state ρ_{0} can be characterized by 3 parameters, we need to obtain the values for 11 × 3 + 3 = 36 parameters. We run 3 × 11 × 4 different experimental settings specified by \(\langle \langle {E}_{0} {R}_{{F}_{k}}{R}_{{G}_{j}}{R}_{{F}_{i}} {\rho }_{0}\rangle \rangle\) with repetitions of 10,000 per setting to collect experimental data \({\bar{m}}_{ijk}\), where i = 1, …, 4 for state preparation, j = 1, …, 11, and k = 1, 2, 3 for different measurement settings. The ansatz prediction \({m}_{ijk}=\langle \langle {E}_{0} {N}_{{F}_{k}}{R}_{{F}_{k}}^{{\rm{id}}}{N}_{{G}_{j}}{R}_{{G}_{j}}^{{\rm{id}}}{N}_{{F}_{i}}{R}_{{F}_{i}}^{{\rm{id}}} {\rho }_{0}\rangle \rangle\) contain Pauli error rates as variational parameters, which we numerically optimize to maximize the likelihood function in Eq. (1). The obtained PTMs are shown in Fig. 2b and Supplementary Fig. 1a.
Once we get experimental PTMs for singlequbit operations, we can derive the inverse of PTM of the noise operation as \({N}_{\mathrm{{s}}}^{1}={R}_{{G}_{\mathrm{{s}}}}^{{\rm{id}}}{R}_{{G}_{\mathrm{{s}}}}^{1}\), which can be decomposed by the combination of PTMs of experimental Pauli operations with \({N}_{\mathrm{{s}}}^{1}={q}_{s,0}{R}_{I}+{q}_{s,1}{R}_{X}+{q}_{s,2}{R}_{Y}+{q}_{s,3}{R}_{Z}\). Then, the ideal operation can be decomposed by experimental operations as \({R}_{{G}_{\mathrm{{s}}}}^{{\rm{id}}}={q}_{s,0}{R}_{I}{R}_{{G}_{\mathrm{{s}}}}+{q}_{s,1}{R}_{X}{R}_{{G}_{\mathrm{{s}}}}+{q}_{s,2}{R}_{Y}{R}_{{G}_{\mathrm{{s}}}}+{q}_{s,3}{R}_{Z}{R}_{{G}_{\mathrm{{s}}}}.\)
Characterization of the twoqubit gate set
The twoqubit gate set, i.e., \({{\mathcal{G}}}_{2}\) includes singlequbit operations in \({{\mathcal{G}}}_{1}^{\otimes 2}\) and twoqubit operations {MS_{YY} and MS_{ZZ}}. As infidelities for the singlequbit gates are about an order lower than those of the twoqubit gates, it is reasonable to divide the maximumlikelihood estimation into two steps.
First, we treat each qubit in the twoqubit system as a singlequbit system and characterize the singlequbit gate set \({{\mathcal{G}}}_{1}\) by gate set tomography, obtaining singlequbit PTMs. The twoqubit PTMs of the singlequbit operations in \({{\mathcal{G}}}_{1}^{\otimes 2}\) is then obtained by a direct product of the singlequbit PTMs on both qubits. As the fiducial set \({{\mathcal{F}}}_{2}\in {{\mathcal{G}}}_{1}^{\otimes 2}\), the PTMs of the fiducial operations are determined at this step.
Second, we characterize the native twoqubit MS_{YY} gate. Under the Pauli error assumption, the PTM of the experimental MS_{YY} gate is decomposed as \({R}_{{{\rm{MS}}}_{YY}}={N}_{{{\rm{MS}}}_{YY}}{R}_{{{\rm{MS}}}_{YY}}^{{\rm{id}}}\), where \({N}_{{{\rm{MS}}}_{YY}}\) is the PTM of the Pauli error channel containing 16 twoqubit Pauli components. After considering the tracepreserving constraint, \({N}_{{{\rm{MS}}}_{YY}}\) has 15 parameters, which are determined by linear equations connecting the ansatz predition \(\langle \langle {E}_{0}^{\otimes 2} {R}_{{F}_{k}}{N}_{{{\rm{MS}}}_{YY}}{R}_{{{\rm{MS}}}_{YY}}^{{\rm{id}}}{R}_{{F}_{i}} {\rho }_{0}^{(1)}\otimes {\rho }_{0}^{(2)}\rangle \rangle\) and corresponding experimental results. To minimize the projection error, we choose 15 linearly independent equations out of 16 × 9 different settings, with most of the measured probabilities close to 0 or 1. Supplementary Fig. 3 shows the corresponding circuits for the experimental settings.
As the MS_{ZZ} is implemented by a MS_{YY} gate sandwiched by proper singlequbit gates, the PTM of the experimental MS_{ZZ} gate is obtained by multiplying the PTMs of the corresponding experimental operations, i.e., \({R}_{{{\rm{MS}}}_{ZZ}}={R}_{{X}_{\frac{\pi }{2}}\otimes {X}_{\frac{\pi }{2}}}{R}_{{{\rm{MS}}}_{YY}}{R}_{{X}_{\frac{\pi }{2}}\otimes {X}_{\frac{\pi }{2}}}\).
Probabilistic errorcancellation scheme
The concrete procedure of applying the probabilistic error cancellation to a given quantum computation task consists of the socalled characterization and calculation phases. The characterization phase is described above. In the calculation phase, we estimate expectation values of quantum circuits with the characterized imperfect quantum device. We first write down the unbiased estimator of the expectation value of a specific quantum circuit as \(\langle \langle {E}_{0}^{{\rm{id}}} {R}_{{G}_{{a}_{L}}}^{{\rm{id}}}\ldots {R}_{{G}_{{a}_{1}}}^{{\rm{id}}} {\rho }_{0}^{{\rm{id}}}\rangle \rangle\), which can be expanded with the quasiprobability distributions obtained in the characterization phase as follows,
where the expection value of \(\langle \langle {E}_{0}^{{\rm{id}}} {R}_{{P}_{{b}_{l}}}{R}_{{G}_{{a}_{L}}}\ldots {R}_{{P}_{{b}_{1}}}{R}_{{G}_{{a}_{1}}} {\rho }_{i}\rangle \rangle\) can be obtained by repeating the specific experimental setting and averaging the measurement results. The straightforward way to evaluate the unbiased estimator is summing over all possible settings. However, this is impractical, because the number of settings grows exponentially with the circuit depth. To alleviate the exponential growth, we rewrite the above expansion as a probability distribution as follows,
with the shorthand notations \({\bf{a}}\equiv \left({a}_{1},\ldots ,{a}_{L}\right)\) and \({\bf{b}}\equiv \left({b}_{1},\ldots ,{b}_{L}\right)\), where \({C}_{{\bf{a}}}\equiv {\sum }_{i,{\bf{b}}}{q}_{0,i}{\prod }_{l}{q}_{{a}_{l},{b}_{l}}\) is the rescaling factor, \({P}_{{\bf{a}}}\left(i,{\bf{b}}\right)={q}_{0,i}{\prod }_{l}{q}_{{a}_{l},{b}_{l}}/C\) is the probability distribution, and \(g(i,{\bf{a}},{\bf{b}})={\rm{sgn}}({q}_{0,i}{\prod }_{l}{q}_{{a}_{l},{b}_{l}})\) is the sign of the setting. Then, we use important sampling to generate M experimental settings, specified by \(\left({i}_{m},{{\bf{b}}}_{m}\right)\) with m = 1, …, M, according to the probability distribution \({P}_{{\bf{a}}}\left(i,{\bf{b}}\right)\), and calculate the expectation value as follows,
where \(O\left({i}_{m},{\bf{a}},{{\bf{b}}}_{m}\right)\) is the result of the projective measurement of the mth setting, being either 0 or 1 in our experiment.
Simple example
In this section, we provide an illustrative example of applying the probabilistic errorcancellation technique to a simple quantum circuit. Suppose an experimenter plans to apply an ideal gate \({G}^{{\rm{id}}}\equiv [{e}^{iY\frac{\pi }{4}}]\) on an ideal initial state \({\rho }^{{\rm{id}}}\equiv \left0\right\rangle \left\langle 0\right\) and get the ideal expectation value of observable \({\left\langle X\right\rangle }^{{\rm{id}}}\equiv Tr[X{G}^{{\rm{id}}}({\rho }^{{\rm{id}}})]=1\). However, as an example of a noisy quantum device, the actual initial state could be \(\rho =90 \% \left0\right\rangle \left\langle 0\right+10 \% \frac{I}{2}\) and the actual gate could be G = 80%G^{id} + 20%D, where \(D(\rho )=\frac{I}{2}\). Then, the actual result is \(\left\langle X\right\rangle =Tr[XG(\rho )]=72 \%\). With the errorcancellation procedure, the ideal initial state is decomposed as \({\rho }^{{\rm{id}}}=(\rho 10 \% \frac{I}{2})/90 \%\) and the ideal gate is decomposed as G^{id} = (G − 20%D) ∕ 80%. Then, the ideal expectation value can be obtained by \({\left\langle X\right\rangle }^{{\rm{id}}}=Tr[XG(\rho )]\times (1/72 \% )Tr[XG(\frac{I}{2})]\times (10 \% /72 \% )\, Tr[XD(\rho )]\times (20 \% /72 \% )Tr[XD(\frac{I}{2})]\times (2 \% /72 \% )\), where the four terms can be obtained by running the noisy quantum device. By computing each term on the noisy quantum device and substituting results into the formula, we can obtain the ideal expectation value.
For a computation with multiple gates, the state preparation, measurement, and each gate can be treated in a similar way. Then, the formula of the ideal expectation value, i.e., a weighted summation of noisy computations has exponential terms with respect to the gate number. Therefore, instead of evaluating each term, we compute the summation using the MonteCarlo method.
In this example, we consider the depolarizing error model. The decomposition can be applied to general error models without correlations. The decomposition formula is obtained by inverting the noise. For the gate G, the noise is N = 80%[I] + 20%D, and G = NG^{id}. The inverse of the noise is N^{−1} = ([I] − 20%D) ∕ 80%. Then, the ideal gate G^{id} = N^{−1}G = (G − 20%D) ∕ 80%.
Analysis on residual errors
Theoretically, the error mitigation technique, combining probabilistic error cancellation and gate set tomography, is capable of completely rectifying the effect of errors in the estimation of expectation values. However, in our experiment, the effective error rates after error mitigation are (1.44 ± 5.28) × 10^{−5} and (0.96 ± 0.10) × 10^{−3} in the singlequbit and twoqubit cases, respectively. Generally speaking, the reasons for the residual errors include the Pauli error assumption, timecorrelated systematic drift, and crosstalk errors between qubits.
In the singlequbit case, the residual errors mainly come from the introduction of the Pauli error model. To quantify the nonPauli error rate, we simulate the dynamics of the same random sequences as those used in the experiment with the characterized experimental PTMs, which are obtained under the Pauli error assumption. The experimental and simulated data of average fidelity are shown in Supplementary Fig. 2a, which are then numerically fitted to extract the error rates. The difference between the simulated and experimental error rates for singlequbit gates is 1.41 × 10^{−5}, which are of the same order of the residual error rate in the singlequbit case. Meanwhile, the data show that the timecorrelated systematic drift has negligible effect and cannot be faithfully quantified within experimental and fitting errors.
In our experiment, we implement two different twoqubit gates, i.e., MS_{YY} and MS_{ZZ} gates. To quantify the residual errors from the Pauli error assumption, we compare the dynamics of the simulated and experimental random twoqubit sequence, where the simulation is based on the characterized PTMs with the Pauli error assumption. The difference between the simulated and experimental error rates gives the estimation of the nonPauli residual error rate, which is about 0.20 × 10^{−3}. As to the crosstalk errors, the situations for MS_{YY} and MS_{ZZ} gates are quite different because of different implementation schemes. Specifically, a MS_{ZZ} gate is implemented by a MS_{YY} gate sandwiched by proper singlequbit gates, which introduce qubitcrosstalk errors. We model the crosstalk effect by measuring an effective Rabi frequency Ω_{eff} on the neighboring ion induced by leakage laser intensities when a singlequbit gate is being implemented by lasers focused on one of the ions. The ratio Ω_{eff} ∕ Ω, with Ω being the Rabi frequency of the target ion, thus quantifies the severity of crosstalk errors. As shown in Supplementary Fig. 4a, we numerically simulate the state fidelities of the original and errormitigated MS_{YY} and MS_{ZZ} gates. As expected, the numerical results show that MS_{YY} gates, either original or errormitigated ones, are insensitive to the crosstalk errors, whereas the fidelities of MS_{ZZ} gates degrade as the severity of crosstalk errors increases. According to the numerical results, the crosstalk residual error rate is about 0.68 × 10^{−3} at the experimental level of qubit crosstalk. Finally, the remaining part of the residual error rate, 0.08 × 10^{−3}, is attributed to the timecorrelated systematic drift.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work was supported by the National Key Research and Development Program of China under Grant numbers 2016YFA0301900 and 2016YFA0301901, and the National Natural Science Foundation of China Grant numbers 11574002, 11504197, 11875050, and 11974200. Y.L. also acknowledges the support by NSAF Grant number U1930403.
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S.Z., Y. Lu, K.Z. and W.C. developed the experimental system. Y. Li proposed theoretical frame of the work. S.Z. and J.N.Z. investigated the actual schemes for the experimental realization. S.Z. led the date taking. K.K. supervised the project. S.Z., Y. Li, J.N.Z. and K.K. wrote the manuscript.
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Zhang, S., Lu, Y., Zhang, K. et al. Errormitigated quantum gates exceeding physical fidelities in a trappedion system. Nat Commun 11, 587 (2020). https://doi.org/10.1038/s4146702014376z
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DOI: https://doi.org/10.1038/s4146702014376z
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