Controlling error orientation to improve quantum algorithm success rates

The success probability of a quantum algorithm constructed from noisy quantum gates cannot be accurately predicted from single-parameter metrics that compare noisy and ideal gates. We illustrate this concept by examining a system with coherent errors and comparing algorithm success rates for different choices of two-qubit gates that are constructed from composite pulse sequences, where the residual gate errors are related by a unitary transformation. As a result, all of the sequences have the same error relative to the ideal gate under any distance measure that is invariant under unitary transformations. However, the circuit success can vary dramatically by choosing error orientations that do not affect the final outcome and error orientations that cancel between conjugate controlled-nots, as demonstrated here with Clifford circuits, compiled Toffoli gates, and quantum simulation algorithms. The results point to the utility of both minimizing the error and optimizing the error direction and also to the advantages of using multiple control sequences for the same gate type within a single algorithm.

Figure 1

A visual of the SK1 pulse sequence. The dominant residual error points orthogonal to the plane of correction (in this figure $a_1$–$a_2$). This second-order rotation has an angle of rotation proportional to the pulse area, $\beta =4{\pi }^{2}sin\left({\varphi }^{SK1}\right)cos\left({\varphi }^{SK1}\right)$.

Figure 2

A circuit for implementing the Bernstein-Vazarani for $f\left(\mathbf{x}\right)=\mathbf{a}·\mathbf{x}$ given $\mathbf{a}=1111$.

Figure 3

Comparison of $1\text{−}{\mathcal{F}}_{\mathrm{circuit}}$ for the four-bit Bernstein-Vazarani circuit for different CNOT implementations as a function of the systematic error $\epsilon$ (solid lines). The underlying $1\text{−}{\mathcal{F}}_{\mathrm{gate}}$ are plotted for the different implementations of CNOT (dashed lines). Although the corrected $\mathrm{CNOTs}({\mathrm{CNOT}}_{XI}$ and ${\mathrm{CNOT}}_{YI}$, which are both examples of ${\mathrm{CNOT}}_{\mathrm{SK}1}$) have the same ${\mathcal{F}}_{\mathrm{gate}}$, they result in drastically different ${\mathcal{F}}_{\mathrm{circuit}}$.

Figure 4

A circuit for implementing the Toffoli gate from single-qubit gates and CNOT gates. Note that the six CNOT gates come in three conjugate pairs.

Figure 5

Comparison of of 1-${\mathcal{F}}_{\mathrm{gate}}$ of the Toffoli gate as a function of the systematic error $\epsilon$ for different CNOT pulse sequences (solid lines). The underlying gate errors of the CNOT sequences making up the Toffoli (Fig. 4) are also presented (dashed lines). Corrected pulse sequences improve gate fidelity for small $\epsilon$ over the naive circuit, but a strategy of canceling errors on the target qubit by conjugate pairs yields the lowest loss in Toffoli gate fidelity and a fundamentally improved scaling.

Figure 6

The phase estimation circuit, which simulates the Hamiltonian $XX+YY$. There is a symmetry for all of the CNOT gates within this circuit. All of the CNOT gates in this circuit have a neighboring CNOT gate, which acts on the same control such that no unitary acts on the control qubit in between them. By ensuring the implementation of these neighboring CNOT pairs are the conjugate of each other with the dominant error placed on the control qubit, the dominant residual error from the CNOT pair will become identity. This technique can be applied to any controlled rotation. The state $|\psi \rangle$ on which phase estimation is performed was chosen as the ground state of $H,\frac{|10\rangle -|01\rangle }{\sqrt{2}}$.

Figure 7

Comparison of $1\text{−}{\mathcal{F}}_{\mathrm{circuit}}$ for the quantum simulation circuit (Fig. 6) for different CNOT gates versus the systematic error parameter $\epsilon$ (solid lines). The underlying CNOT errors, $1\text{−}{\mathcal{F}}_{\text{gate}}$, for SK1 corrected and uncorrected CNOT gates are also plotted (dashed) lines. The best error suppression occurs when the corrected gates are chosen as conjugate pairs (${\mathrm{CNOT}}_{\pm XI}$).