Abstract
Medium-scale quantum devices that integrate about hundreds of physical qubits are likely to be developed in the near future. However, these devices will lack the resources for realizing quantum fault tolerance. Therefore, the main challenge of exploring the advantage of quantum computation is to minimize the impact of device and control imperfections without complete logical encoding. Quantum error mitigation is a solution satisfying the requirement. Here, we demonstrate an error mitigation protocol based on gate set tomography and quasi-probability decomposition. One- and two-qubit circuits are tested on a superconducting device, and computation errors are successfully suppressed. Because this protocol is universal for digital quantum computers and algorithms computing expected values, our results suggest that error mitigation can be an essential component of near-future quantum computation.
INTRODUCTION
Quantum computers are quantum-mechanical devices capable of solving problems that are believed to be intractable for classical computers. The most essential issue in practicing quantum computation is to deal with imperfections of the device and control that cause computation errors. Quantum error correction can suppress the chance of errors to an arbitrarily low level, which, however, is beyond the scope of near-future technologies (1, 2). For shallow algorithms executed on near-future quantum devices (3–9), quantum error mitigation methods (10, 11) have been proposed recently to attain a computation result with minimal errors. This approach is practical because complete logical encoding is not required. The probabilistic error cancellation (PEC) protocol based on a combination of gate set tomography (GST) and quasi-probability decomposition is one of those methods (12), which can be applied to any platform without prior knowledge of imperfections and works for any algorithm that outcomes are expected values of certain observables. In this method, certain quantum circuits are first executed to identify a model of imperfections, and then random circuits are sampled from a distribution according to the model (see Fig. 1A). The theory suggests that the average of these random circuits can provide an accurate computation result. Here, we demonstrate the experimental realization of this method for the first time. The device is a superconducting circuit consisting of 10 frequency-tunable transmon qubits, among which four qubits are actively used in the demonstration. Details of the device can be seen in (13). We use single- and two-qubit circuits to test this method and find that, with error mitigation, the computation accuracy is substantially improved.
(A) Flowchart of the universal quantum error mitigation, which has two stages: GST and the random circuit computation. (B) Layout of four qubits actively used in the experiment. The information is encoded in Q1 and Q2, and the other two qubits QA1 and QA2 are ancillary qubits. (C) Controlled-ϕ-phase gate Cϕ realized using the dressed state gate Uϕ and single-qubit gates. The single-qubit gate
We use GST (14–16) to acquire information about the measurement and gate errors in the experiment, which is then used in PEC to decompose any ideal measurement or gate by those experimentally accessible ones with errors. GST can be seen as a self-consistent extension of the quantum process tomography, which takes into account all the errors that occurred in the experimental operations including state preparations, quantum gates, and measurements.
We use Pauli transfer matrix (PTM) representation to notate quantum states, quantum gates, and measurements as commonly adopted in quantum tomography. We define σi as the ith operator from the n-qubit Pauli basis
(A) Circuits for one- and two-qubit GST. The gate to be characterized (marked in gray) is implemented in between the state preparation and measurement. Gram matrices and matrices of measurement-initialization gates are obtained using the one-qubit circuit. Matrices of two-qubit gates are obtained using the two-qubit circuit. For the gram matrix, the gate is null. (B) Gram matrix gexp of the qubit Q1. (C) PTMs of the two-qubit gate Cπ. For the ideal gate, each element is calculated as
Experimental operations of Aexp, Bexp, and
Given the decomposition formulas (12), we randomly generate circuits modified from the original circuit of the computation task and implement these random circuits to obtain the computation result with error mitigated, because errors in random circuits cancel each other when taking the average. Here, we only decompose and replace the measurement and two-qubit gate in the original circuit while the state preparation and single-qubit gates are unchanged. We do this because the error of the previous two is larger than the latter by an order of magnitude. To be explicit, by heralding the ground state for qubit preparation and measuring the heating rate (17), we estimate the ground state preparation error to be below 0.25%; error rates of single-qubit gates are calibrated to be below 0.25% in randomized benchmarking (18–20). In comparison, the readout error is about 3.5% for the ground state and 5.7% for the excited state; the two-qubit gates applied in this paper have errors around 7%. Gate fidelities can be further boosted on our device, however, this is unnecessary for the purpose of PEC demonstration.
RESULTS
Mitigating readout error in one-qubit computation
We first test the effect of PEC with a one-qubit computation, whose circuit is shown in Fig. 3A. In this circuit, we initialize the qubit by heralding the state ∣0〉, with a state fidelity above 0.997. Then, the gate
(A) Circuit of the one-qubit computation. In PEC, the measurement of the observable Z is replaced by random gates. (B) Random circuit of the one-qubit computation, in which the measurement in the original circuit is replaced by the measurement of the observable σi. (C) Circuit of the two-qubit DQCp computation. In PEC, the two-qubit gate and the measurement are replaced by random operations. (D) Circuit of the Pauli twirling. (E) Representative random circuits of the two-qubit computation. μ denotes the outcome of the corresponding measurement, and w is the weight of the corresponding circuit wj,i as defined in the main text (in the figure, the subscript of w denotes the number of the instance). N is the total number of instances. The circuit in the blue box is the replacement of the two-qubit gate Cϕ. We note that, in instance 1, four single-qubit gates are Pauli gates of the Pauli twirling. (F) Results of the one-qubit computation. The probability distribution of the computation result is plotted. Without error, the ideal result is 〈Z〉 = 0. (G) Results of the two-qubit computation. Each data point is obtained using 1,000,000 instances. We implement random circuits 10,000 times to compute one average value of X and repeat the computation to obtain 100 average values. The error bar indicates the SD of these average values.
Because the state preparation and single-qubit gate are both quite precise, the relatively large deviation of 〈Z〉exp from zero, i.e., the ideal result, is mainly due to the readout error, which we intend to mitigate by decomposing the readout operation. To work out the decomposition formula, we obtain the Gram matrix gexp in the experiment, and the result is shown in Fig. 2B. Assuming the error-free state preparation, we can take a reasonable estimate of Aexp as
Mitigating readout and entangling gate errors in two-qubit computation
Now, we turn to a two-qubit computation, taking the deterministic quantum computation with pure states (DQCp) (21) as an example. The circuit is shown in Fig. 3C. A main error source in this circuit is the controlled-ϕ-phase gate
To mitigate the error in Cϕ, we need first to work out the decomposition formula. Given the ideal superoperator U representing Cϕ, the decomposition formula reads
In our experiment, 257 operations are used for decomposing an ideal Cϕ gate. The first 256 operations are generated from the tensor product of 16 single-qubit operations, which include measurement and reset gates, as listed in Table 1. The 257th operation is the gate Cϕ modified by the Pauli twirling as explained in Materials and Methods. We reconstruct the experimental operations of Cϕ and single-qubit measurement-reset gates in GST, while we simply assume that single-qubit gates are error-free, because single-qubit gates can be experimentally implemented with high fidelity. Unlike the state preparation, assuming error-free single-qubit gates can potentially cause inaccuracy in the quantum computation with PEC. PTM of the controlled-π-phase gate Cπ obtained using GST is illustrated in Fig. 2C as an example. These 257 operations are linearly independent, which ensures that the decomposition solutions can always be found by solving a system of linear equations. In all solutions, we choose the one with the minimum in
The measurement-reset operation is occasionally used in random circuits (see Table 1). To minimize the time of reset, we realize the reset gate using ancilla qubits QA1 and QA2 on the same chip (see Fig. 1B). Each reset operation uses an ancilla qubit initially prepared in the ground state ∣0〉, and then a swap gate is applied to reinitialize the target qubit when the reset is requested (13), following which a single-qubit gate G with the fidelity above 0.997 rotates the qubit to the state ∣ψ〉, as shown in Fig. 1D. The whole measurement-reset operation typically has a fidelity of around 0.916.
In the two-qubit computation with PEC, to mitigate both readout and two-qubit gate errors, the gate Cϕ is randomly replaced by the gate
In the experiment, we adjust the phase ϕ of Cϕ and measure the expected value of X. When implementing the computation with PEC, we randomly sample a circuit according to the decompositions of both Cϕ and the observable X. Representative decomposed sampling circuits are shown in Fig. 3E. The experiment result is shown in Fig. 3G, which demonstrates a substantial improvement on the computation accuracy.
The most substantial improvement is obtained at ϕ = π/2, in which case the difference between the computation result and the ideal value is reduced from 0.1690 to 0.0102 by using PEC. To estimate the fidelity required to achieve the same computation accuracy, we consider a quantum system with depolarizing error channels (23) and assume that the state preparation and single-qubit gates are ideal. The depolarizing error channel either preserves or completely destroys the information with certain probabilities (see the Supplementary Materials), which does not characterize our device. We choose the depolarizing model because it takes all possible errors into account with equal probability. In the depolarizing model, the two-qubit gate and measurement with the fidelity ∼99.3% are required to achieve the computation accuracy 0.0102, which is comparable to the highest fidelity reported in the superconducting qubit system (20, 24).
DISCUSSION
For multiqubit devices, GST of the entire device is not practical, because the experiment time increases exponentially with the qubit number. Similarly, the number of operations for decomposing a multiqubit gate also increases exponentially with the qubit number. Single- and two-qubit gates are sufficient for the universal quantum computation. Therefore, if errors are uncorrelated, then we only need to implement GST up to two qubits and decompose two-qubit gates, as demonstrated in our experiment. Errors are uncorrelated if the evolution of two qubits under a two-qubit gate is independent from the evolution of other qubits. As a result of the evolution, the quantum operation on the entire device can be factorized into the product of an operation on the two qubits and operations on other qubits. It is similar for single-qubit gates. In our device, the primary dephasing noise is dominated by fluctuators in the form of magnetic moments, whose influence is local in each individual physical qubit, and therefore, the dephasing-induced errors are uncorrelated between qubits. In our experiment, we have neglected error correlations in GST so that single-qubit operations are characterized in single-qubit tomography even in the two-qubit experiment. Neglecting error correlations sacrifices accuracy of PEC. A notable effect of correlations on the computation result is not observed in our experiment.
We have experimentally demonstrated that PEC, a universal quantum error mitigation protocol, can substantially reduce the error in quantum computation on a noisy quantum device. The protocol in our experiment does not require subthreshold error rate or tremendous additional physical qubit resources. Compared with the algorithm-specified protocol (7) and the extrapolation of gate time (9), the combination of GST and quasi-probability decomposition is not restricted to the algorithm or error model. A few techniques in PEC are explored: estimate of the state preparation matrix according to ideal states, Pauli twirling for randomizing the error, approximate GST, and decomposition neglecting error correlations. An important factor limiting the circuit depth in our demonstration is the variance of computation result, which depends on the error rate of quantum gates. Improvement in gate fidelity can extend the circuit depth, and relatively deep circuits can be implemented on intermediate-scale devices with a feasible fidelity (12, 25). By demonstrating the power of error mitigation techniques on the superconducting quantum device, our results highlight the potential of using such techniques in computation tasks on near-future quantum devices.
MATERIALS AND METHODS
Our device is a superconducting circuit consisting of 10 frequency-tunable transmon qubits, among which four qubits are actively used in the demonstration. Details of the device can be seen in (13).
Pauli twirling converts the error in a gate into stochastic Pauli errors (26–28), which can reduce the variance of the computation result (12). The circuit of the gate Cϕ with Pauli twirling is shown in Fig. 3D. We sandwich Cϕ in between four Pauli gates (two for each qubit), which are randomly chosen but conserve the gate Cϕ up to a global phase difference. If all gates are error free, then the two-qubit gate realized in this way is still Cϕ, i.e., [Cϕ] = ∑a,bpa,b[σc ⊗ σdCϕσa ⊗ σb]. Here, we used the bracket notation to denote a superoperator [C](ρ) = CρC†, {σi} are single-qubit Pauli gates chosen to satisfy σc ⊗ σdCϕσa ⊗ σb = ηCϕ, η can be any phase factor, and pa,b is the probability. If gates have errors, then the two-qubit gate will be effectively changed by the Pauli twirling. Using the twirled gate as the 257th operation in the decomposition, we need the estimate of the twirled gate, which is
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/9/eaaw5686/DC1
Randomized benchmarking for single-qubit gates
Heating rate measurement
Readout error for Q1 and Q2
One-qubit QEM experiment
Measurement and reset gates
Decomposing Cϕ gate
Depolarizing error channels
Fig. S1. Randomized benchmarking data.
Fig. S2. Heating rate measurement.
Fig. S3. One-qubit QEM experiment.
Fig. S4. Measurement-reset gate.
Fig. S5. Decomposition of the Cπ gate.
Table S1. Error rates of readout measured by repeatedly preparing the state ∣0〉 or ∣1〉 and measuring the probability of incorrect output.
Table S2. Gate fidelities for all measurement-reset gates used in this paper.
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REFERENCES AND NOTES
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