Quantum computation with universal error mitigation on a superconducting quantum processor

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 Xπ/2=e−iπ4X, whose gate fidelity is calibrated to be 0.998 by randomized benchmarking, is applied to rotate the qubit state around the x axis by π/2, after which the operator Z is measured. The measurement is denoted by M_Z in the circuit, which yields an outcome of −1 or 1. We repetitively implement the circuit 3000 times and take the average as the expected value of Z. We then repeat the same procedure to obtain 100 expected values. Sample numbers are the same in non-PEC and PEC experiments. The histogram analysis of the resulting data is shown in the top panel of Fig. 3F. The average over 100 expected values gives 〈Z〉^exp = 0.027 ± 0.016.

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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 g^exp in the experiment, and the result is shown in Fig. 2B. Assuming the error-free state preparation, we can take a reasonable estimate of A^exp asÂ=(11110010000−11−100)where the Pauli operator basis is sorted in the order I, X, Y, and Z. With B̂=gexpÂ−1, we can decompose the observable Z in the form Z=∑iqiσ̂i, where 〈〈σ̂i∣ are rows of B̂, and the quasi-probabilities q_i are elements of the vector q=〈〈Z∣B̂−1. Quasi-probabilities are real but can be greater than unity and be negative. In the quantum computation with PEC, we implement random circuits 3000 times. In each of them, the measurement of Z in the original circuit is replaced by the measurement of σiexp (see Fig. 3B) with the probability ∣q_i∣/∑_l ∣q_l∣. In the experiment, measurements of σ1exp and σ2exp, i.e., X- and Y-basis measurements, are implemented by inserting gates Y_π/2 and X_π/2, respectively, before the usual Z-basis measurement. Here, σiexp is the physical measurement whose estimate is σ̂i. The measurement outcome is +1 or −1 (the outcome is always +1 in the measurement of σ0exp, i.e., the identity operator I). When taking the average of measurement outcomes, each outcome is multiplied by a weight factor of w_i = sgn(q_i) ⋅ ∑_l ∣q_l∣. We take this weighted average as the expected value of Z. Then, we repeat the above procedure to obtain 100 expected values, and the average over all these expected values yields 〈Z〉^PEC = 0.003 ± 0.022 as shown in the bottom panel of Fig. 3F. The accuracy of the computation is successfully improved compared with the computation without PEC.

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 Cϕ=I+Z2⊗I+I−Z2⊗Zϕ, where Zϕ=e−iϕ2Z. This gate is realized with a two-qubit dressed state gate U_ϕ (22) plus 10 single-qubit gates, as shown in Fig. 1C. The two-qubit dressed state gate essentially achieves a controlled-ϕ-phase gate in the X basis, and single-qubit gates are used to transform the X basis into the Z basis. Fidelities of different C_ϕ gates are 0.958 ± 0.010, 0.935 ± 0.011, 0.920 ± 0.011, and 0.915 ± 0.011 for ϕ = π/4, π/2, 3π/4, and π, respectively, characterized using GST. Gate fidelities can be further boosted on our device, however, this is unnecessary for the purpose of PEC demonstration.

To mitigate the error in C_ϕ, we need first to work out the decomposition formula. Given the ideal superoperator U representing C_ϕ, the decomposition formula reads U=∑jqj′Ûj, where qj′ is a set of quasi-probabilities as introduced previously and Ûj is the estimate of a set of experimentally achieved operations Ujexp, as defined below and in Table 1.

In our experiment, 257 operations are used for decomposing an ideal C_ϕ 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 C_ϕ modified by the Pauli twirling as explained in Materials and Methods. We reconstruct the experimental operations of C_ϕ 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 C_π 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 ∑k∣qk′∣ to minimize the variance of the computation result.

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 C_ϕ is randomly replaced by the gate Ujexp with the probability ∣qj′∣/∑k∣qk′∣, and the measurement of X is replaced by the measurement of σiexp with the probability ∣qi′′∣/∑l∣ql′′∣, where q′′=〈〈X∣B̂−1. Similar to the one-qubit case, the measurement outcome of each random circuit is multiplied by a weight factor of wj,i=sgn(qj′qi′′)⋅∑k∣qk′∣⋅∑l∣ql′′∣, and we take the weighted average as the computation result, as shown in Fig. 3E.

In the experiment, we adjust the phase ϕ of C_ϕ and measure the expected value of X. When implementing the computation with PEC, we randomly sample a circuit according to the decompositions of both C_ϕ 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).