Systematic Magnus-Based Approach for Suppressing Leakage and Nonadiabatic Errors in Quantum Dynamics

We present a systematic, perturbative method for correcting quantum gates to suppress errors that take the target system out of a chosen subspace. Our method addresses the generic problem of nonadiabatic errors in adiabatic evolution and state preparation, as well as general leakage errors due to spurious couplings to undesirable states. The method is based on the Magnus expansion: By correcting control pulses, we modify the Magnus expansion of an initially given, imperfect unitary in such a way that the desired evolution is obtained. Applications to adiabatic quantum state transfer, superconducting qubits, and generalized Landau-Zener problems are discussed.

Popular Summary

Quantum computing, which relies on bizarre behaviors of atomic particles such as superposition and entanglement, has the potential to speed up complex computational problems. Unlike a traditional computer, which processes information using digital bits (0s and 1s), quantum computers rely on quantum bits (qubits). Manipulating the information stored in a quantum bit can be tricky, as a quantum bit can have more than two logical states. We have developed a method to ensure that at the end of any qubit manipulation, the information remains encoded in the two states representing the logical 0 and 1.

We show how to implement protocols to manipulate information while remaining in the qubit logical subspace. Our solution is analogous to carrying a tray of drinks in a timely manner. Speeding up the pace will get the drinks out faster, but the drinks might spill. To avoid spilling, the host can constantly tilt the tray to keep liquid from sloshing out of the glasses. In quantum technologies, we would like to bring a system from state A to state B. However, when performing this operation as quickly as possible, the system often ends in some undesired state C. Our technique describes a way to dynamically “tilt” the quantum dynamics—by manipulating an external magnetic field or voltage, for example—to arrive at B quickly and reliably.

Our method could lead to increased fidelity for a variety of quantum control protocols, not just in the general area of quantum computing but also, for example, in quantum sensing and quantum simulators. Future refinements to our technique could include the effects of noise.

Figure 1

Schematic representation of the Magnus-based algorithm. (a) Before applying the algorithm, the ideal evolution is corrupted by the spurious coupling, which prevents it from reaching the target state $|{\psi }_{\mathrm{f}}⟩$. (b) After the first step of the Magnus-based algorithm, “pure” leakage errors are canceled on average by the control Hamiltonian ${\widehat{W}}_1(t)$. Although the system ends in ${\mathcal{H}}_{\mathrm{comp}}$ at $t_{\mathrm{f}}$, the target state $|{\psi }_{\mathrm{f}}⟩$ is not reached. (c) The second step of the Magnus-based algorithm corrects induced phase errors within ${\mathcal{H}}_{\mathrm{comp}}$ with the help of the control Hamiltonian ${\widehat{W}}_2(t)$.

Figure 2

Flowchart illustrating the first few steps of the Magnus-based algorithm.

Figure 3

(a) Fidelity error for STIRAP-style adiabatic state transfer in a $\mathrm{\Lambda }$ system [cf. Eq. (35)], as a function of the adiabatic parameter $\nu /G_0$. Different curves correspond to different choices of control corrections: “Uncorrected” control sequence defined in Eq. (36) (black), first-order Magnus corrected controls (green), second-order Magnus corrected controls (blue), and the optimal second-order Magnus corrected controls (red). (b) Modified pump pulse $G_{\mathrm{p}}(t)$ predicted by the second-order Magnus algorithm as a function of $t/t_{\mathrm{f}}$ for different values of $\nu /G_0$. As $\nu /G_0\to 0$, the modified sequence converges to the original one. The Stokes pulse $G_{\mathrm{s}}(t)$ is the time reverse of the pump pulse.

Figure 4

(a) Fidelity error for STIRAP-style state transfer, as a function of adiabatic parameter $\nu /G_0$, for uncorrected pulses having a simple Gaussian form. Curves correspond to different choices of correction: “Uncorrected” control sequence defined in Eq. (44) (black), first-order Magnus-corrected controls (green), and second-order Magnus-corrected controls (blue). (b) Modified pump pulse $G_{\mathrm{p}}(t)$ predicted by the second-order Magnus algorithm as a function of $t/t_{\mathrm{f}}$ for different values of $\nu /G_0$. As $\nu /G_0\to 0$, the modified sequence converges to the original one. The Stokes pulse is the time reverse of the pump pulse. The black dashed line is the control predicted by DRAG for $\nu /G_0=0.475$. As one can observe, this control diverges when $t\to 0$.

Figure 5

Fidelity error in the implementation of a Hadamard gate in a superconducting qubit having an extra leakage level, cf. Eq. (53). We plot the error in the state-averaged infidelity as a function of the adiabatic parameter ${\kappa }_0/\mathrm{\Delta }$. We choose $\lambda =\sqrt{2}$, $-t_{\mathrm{i}}=t_{\mathrm{f}}=3/{\kappa }_0$, and $t_0=\sqrt{\pi }/(4{\kappa }_0)$. Curves correspond to different choices of corrections: Uncorrected Gaussian control pulse (black), first-order DRAG-corrected control as given in Eq. (9) of Ref. [4] (orange), unconstrained Magnus-based control sequence [see Eqs. (59) and (60)] (green), and constrained Magnus-based control sequence [see Eqs. (67) and (68)] (blue).

Figure 6

(a) Adiabatic energy levels of the Landau-Zener-Stückelberg-Majorana model as a function of time. A purely adiabatic evolution realizes an effective NOT gate on the qubit levels $|0⟩$, $|1⟩$. Colors indicate how the adiabatic eigenstates evolve in the $|0⟩$, $|1⟩$ basis. (b) A similar plot of adiabatic energy levels versus time for the multiple-crossings Landau-Zener model, where in addition to the qubit levels, there are two additional leakage levels $|2⟩$, $|3⟩$. Colors again indicate how the adiabatic eigenstate wave functions evolve in the $|0⟩$, $|1⟩$, $|2⟩$, $|3⟩$ basis. Because of mixing with the leakage levels, a pure adiabatic evolution no longer results in a NOT gate of the qubit states. Here, we sketch the situation where ${\omega }_2={\omega }_3=\omega$, the spurious couplings are comparable to $\eta$, and $\omega \lesssim \eta$.

Figure 7

Modulus square of the amplitude coefficient of $|1⟩$ for the ground state of the Landau-Zener and MLZ models. We plot the coefficient as a function of time $\tau$ for $\eta =1$ and ${\omega }_2={\omega }_3=0.1$. For the MLZ model, the value is substantially different from 1 for large values of $\tau$. This indicates that a pure adiabatic evolution will not result in a NOT gate of the qubit states.

Figure 8

Fidelity error for an attempted $|0⟩\to |1⟩$ state transfer in the MLZ model with two leakage levels [cf. Eq. (74)], as a function of the dimensionless coupling $\eta$. We have taken the couplings involving spurious levels to be the same as the coupling $\eta$ between qubit levels, i.e., ${\eta }_{03}={\eta }_{12}={\eta }_{23}=\eta$, and we have set the energy offsets ${\omega }_2={\omega }_3=0.1$. Different curves correspond to different possible corrections (as indicated in the figure). Without any correction, small errors are not possible for any choice of $\eta$; in contrast, if one includes the Magnus corrections, errors less than ${10}^{-3}$ are possible for a wide range of $\eta$.

Figure 9

Real (purple lines) and imaginary (green lines) parts of the controls predicted by the Magnus-based algorithm for the MLZ model as a function of time $\tau$ for $\eta ={\eta }_{12}={\eta }_{03}={\eta }_{23}=0.3$ and ${\omega }_2={\omega }_3=0.1$. (a) Detuning of $|0⟩$. (b) Detuning of $|1⟩$. (c) Coupling $\eta$ between the qubit states $|0⟩$ and $|1⟩$. (d) Coupling between $|1⟩$ and the leakage level $|2⟩$. (e) Coupling between $|1⟩$ and the leakage level $|3⟩$. (f) Coupling between $|0⟩$ and the leakage level $|2⟩$. (g) Coupling between $|0⟩$ and the leakage level $|3⟩$. (h) Coupling between the leakage levels $|2⟩$ and $|3⟩$.

Figure 10

Infidelity for STIRAP using Gaussian pulses. The approximate expression for the infidelity as given in Eq. (e5) agrees with exact numerics in the adiabatic regime, $\nu /G_0<1$.