Gate-error analysis in simulations of quantum computers with transmon qubits

In the model of gate-based quantum computation, the qubits are controlled by a sequence of quantum gates. In superconducting qubit systems, these gates can be implemented by voltage pulses. The success of implementing a particular gate can be expressed by various metrics such as the average gate fidelity, the diamond distance, and the unitarity. We analyze these metrics of gate pulses for a system of two superconducting transmon qubits coupled by a resonator, a system inspired by the architecture of the IBM Quantum Experience. The metrics are obtained by numerical solution of the time-dependent Schrödinger equation of the transmon system. We find that the metrics reflect systematic errors that are most pronounced for echoed cross-resonance gates, but that none of the studied metrics can reliably predict the performance of a gate when used repeatedly in a quantum algorithm.

Figure 1

Pulse sequences for the three different realizations of a cnot gate studied in this paper. Gaussians implement $X_{\pi /2}$ and $X_{\pi }$ rotations, and flat-topped Gaussians represent cross-resonance (CR) pulses (i.e., they oscillate at the frequency ${\overline{\omega }}_T$ of the target qubit). The CR1 gate consists only of flat-topped Gaussian pulses at the target frequency ${\overline{\omega }}_T$. The CR2 gate is an echoed CR gate containing two additional $X_{\pi }$ pulses on the control qubit and one $X_{\pi /2}$ pulse on the target qubit. The CR4 gate is a four-pulse echoed CR gate that contains an additional $X_{\pi }$ pulse on the target qubit. See Fig. 6 in Appendix pp2 for the full pulse specifications.

Figure 2

Evolution of the error rate given by Eq. (13) after $n$ applications of a certain gate. (a) Single-qubit Gaussian derivative pulses defined by Eq. (8); (b)–(d) two-qubit cnot gates based on the cross-resonance effect. While CR1 includes only one CR pulse on each of the transmons, CR2 and CR4 employ additional $X$ gates to echo out certain errors (see Sec. 2 and Appendix pp2).

Figure 3

Statistical distances between the ideal result and the measured distribution of states for the circuit ${\text{cnot}}_{12}^n|\psi \rangle$ with $|\psi \rangle =|00\rangle$ (stars) and $|\psi \rangle =|10\rangle$ (circles). (a) Experimental results on the IBMQX; (b)–(d) simulation results. Generically, the echoed cnot versions show worse performance on state $|00\rangle$ than on state $|10\rangle$, both in the experiment and the simulation. Interestingly, this systematic error is not present in the one-pulse version CR1.

Figure 4

Results for a set of quantum circuits creating and characterizing the singlet state $(|01\rangle -|10\rangle )/\sqrt{2}$ as a function of the angles ${\vartheta }_1$ and ${\vartheta }_2$. $F_{1/2}({\vartheta }_1,{\vartheta }_2)$ are single-qubit averages and $F({\vartheta }_1,{\vartheta }_2)$ is a two-qubit correlation function. The corresponding theoretical expectations are given by $E_{1/2}({\vartheta }_1,{\vartheta }_2)$ and $E({\vartheta }_1,{\vartheta }_2)$. (a) ${\vartheta }_1=0$ fixed and ${\vartheta }_2$ variable; (b) ${\vartheta }_1={\vartheta }_2$ variable. Apart from a small systematic deviation around ${\vartheta }_1={\vartheta }_2=135$, the agreement is almost perfect.

Figure 5

Scan of the CR drive amplitudes ${\mathrm{\Omega }}_{\text{CR}}$ on the control qubit and ${\mathrm{\Omega }}_{\text{cancel}}$ on the target qubit. The dimensionless amplitudes can be converted to the strength of the drive by multiplying them with $b_i=2E_{Ci}{(E_{Ji}/8E_{Ci})}^{1/4}$ (shown on top of the plots). The IX and ZX interaction strengths are inferred by measuring the oscillations of the target qubit conditional on the control qubit being in state $|0\rangle$ and state $|1\rangle$ [13]. The linear theory predictions can be derived perturbatively [22] and are only valid for weak drivings. Note that the additional drive on the target qubit (the two figures on the right, shown for ${\mathrm{\Omega }}_{\text{CR}}=0.1$ fixed) linearly displaces IX only. Thus it can either be tuned to single out ZX or to generate the cnot gate directly up to local Z rotations.

Figure 6

Specifications of the pulse sequences in Fig. 1 to implement a generic cnot gate with VZ phases. The elementary Gaussian pulse GD is defined in Table 4, and GF is defined in Table 5. C (T) is the control (target) qubit. (a) Generic cnot gate with the preceding VZ phases that all pulses need to be capable of shifting through; (b) one-pulse CR1 gate which includes a flat-topped Gaussian pulse on the control and the target qubit simultaneously; (c) two-pulse echoed CR2 gate; (d) four-pulse echoed CR4 gate. In the CR2 and the CR4 scheme, the additional phase shift $\xi$ is zero if ${\overline{\omega }}_C>{\overline{\omega }}_T$ and $\pi$ otherwise to handle the case when ZX is negative (see Fig. 5).

Figure 7

Circuit for the two-qubit QFT.

Figure 8

Circuit for experiments on the singlet state.