Time-Dependent Hamiltonian Reconstruction Using Continuous Weak Measurements

Reconstructing the Hamiltonian of a quantum system is an essential task for characterizing and certifying quantum processors and simulators. Existing techniques either rely on projective measurements of the system before and after coherent time evolution and do not explicitly reconstruct the full time-dependent Hamiltonian or interrupt evolution for tomography. Here, we experimentally demonstrate that an a priori unknown, time-dependent Hamiltonian can be reconstructed from continuous weak measurements concurrent with coherent time evolution in a system of two superconducting transmons coupled by a flux-tunable coupler. In contrast to previous work, our technique does not require interruptions, which would distort the recovered Hamiltonian. We introduce an algorithm, which recovers the Hamiltonian and density matrix from an incomplete set of continuous measurements, and demonstrate that it reliably extracts amplitudes of a variety of single-qubit and entangling two-qubit Hamiltonians. We further demonstrate how this technique reveals deviations from a theoretical control Hamiltonian, which would otherwise be missed by conventional techniques. Our work opens up novel applications for continuous weak measurements, such as studying nonidealities in gates, certifying analog quantum simulators, and performing quantum metrology.

Popular Summary

A contemporary challenge in quantum computing is the development of high-fidelity quantum gates and identification of their limitations. Current characterization techniques use projective measurements, which collapse the quantum state of a system, before and after the qubit to evaluate the quality of quantum gates in a processor but do not directly reveal the dynamics of the qubits. We introduce a new technique for analyzing the evolution of a system of qubits using continuous weak measurements, which allow us to monitor the quantum state without collapsing it, revealing the real-time dynamics of the qubits.

We continuously measure the system while it evolves under a quantum gate. Because the weak measurements do not directly reveal the full state of the qubits, we develop a new method for reconstructing the full state during the evolution from those weak measurements. We then demonstrate in a system of two superconducting qubits that this technique can reveal dynamics that would otherwise be missed by other methods.

With our ability to reveal the dynamics at the level of the qubits, we anticipate that our technique will shed light on what limits gate fidelities and how to improve control methods as well as inspire applications of weak measurements to quantum simulators.

Figure 1

Hamiltonian reconstruction method. (a) To reconstruct an unknown target Hamiltonian $H_t(t)$ of a system composed of two fixed frequency transmons (Q1, Q2) coupled via flux-tunable coupler (C12), we continuously measure the qubits while applying pulses to generate $H_t(t)$. Single-qubit target Hamiltonians are generated by pulses of duration of $t_p$ on the charge lines (not shown). Entangling two-qubit target Hamiltonians are generated by applying rf pulses on the flux line (FF). In the adiabatic limit, the noisy resonator output field [schematically depicted in (b)] follows the evolution of the qubit $z$ coordinate during the pulse. (c) We average the voltage records $V_{1,2}$ over many shots, apply a low-pass filter to the averaged signal, and then deconvolve it to recover the resulting qubits’ $z$ evolution for $S\ge 2$ different initial states while keeping the qubit and coupler pulses fixed. The resulting ensemble averages $z_{1,2}(t)$ along with the calibrated measurement-induced dephasing rate ${\mathrm{\Gamma }}_d$ are inputted to the Hamiltonian reconstruction algorithm. (d) The output of the algorithm is an estimate of the time-dependent target Hamiltonian expanded in the Pauli operator basis.

Figure 2

Single-qubit Hamiltonian reconstruction. (a) Pulse sequence. We weakly measure Q1 while applying a qubit control pulse (duration $t_p$). After the pulse we use state tomography as independent validation of the reconstructed Hamiltonian $H_t(t)$. A 40-ns buffer is added between consecutive pulses to ensure the resonator has reached a steady state. (b) Reconstructed Hamiltonian ($\mathrm{\Delta }t=2\mathrm{ns}$) for three different pulses: (i) Calibrated $\pi$ pulse with flat top and cosine ramp. (ii) Same as in (i) plus additional out-of-phase waveform (iii) Same as in (i) plus a sinusoidal amplitude “error” pulse. (iv) Extracted $\pi$-pulse error estimated from the reconstructed Hamiltonian subtracted from calibrated estimate. (c) Postpulse reconstruction infidelity quantifying overlap between predicted final state and state tomography for the three pulses in (b). (d) Decoupling of resonator from the qubit. For fast pulses, the qubit coordinate $z(t)$ obtained from the resonator evolution (solid lines, transparency indicates evolution after the end of the $\pi$ pulse) deviates from the QuTiP simulation (dashed lines). (e) Reconstruction infidelity averaged over all initial states for different gate durations. Error bars represent spread of $1-{\mathcal{F}}_r$ over different initial states used. The qubit simulation (dashed line), which includes the readout resonator captures the measured trend and shows that the experimental value saturates the minimum for all pulse lengths. Low reconstruction fidelity for fast pulses is explained by the deviation observed in (d).

Figure 3

Two-qubit gate operation. (a) Two fixed-frequency transmon qubits (Q1, Q2) are coupled to a tunable coupler (SQUID). The flux line FF is used to dc bias and modulate the coupler flux ${\mathrm{\Phi }}_{\mathrm{ext}}$. (b) By dc biasing the coupler, the qubit and bus frequencies and couplings can be tuned. The coupler is biased at ${\mathrm{\Phi }}_{\mathrm{ext}}=0$ (marked by the red star) where the qubit frequency [fit to Eq. (30)] is first-order insensitive to the flux (inset). The static $ZZ$ rate $\zeta$ is measured using a JAZZ measurement [74], and the data are fit using models described in Appendix pp6. (c) Population oscillations of the $|10⟩$ state due to driving the flux line (FF) with an ac flux drive at frequency ${\omega }_{\mathrm{\Phi }}$ near $\frac{1}{2}({\omega }_q^{(1)}-{\omega }_q^{(2)})$. (d) Populations of the $|10⟩$ and $|01⟩$ states after preparing the qubits in $|10⟩$ and driving the coupler at ${\omega }_{\mathrm{\Phi }}/2\pi =23.7\mathrm{MHz}$ for various pulse durations. The populations oscillate out of phase, indicating Q1 and Q2 are swapping an excitation. A full swap of an excitation occurs when $\theta =\pi$ [as in Eq. (33)], taking 163 ns.

Figure 4

Two-qubit Hamiltonian reconstruction. (a) Pulse sequence. The coupler rf pulse generates an entangling two-qubit Hamiltonian. (b) We study three different pulses $XY(\beta ,\theta )$ whose ideal action in the $\{|01⟩,|10⟩\}$ subspace is schematically depicted on the Bloch sphere. (c) Reconstructed Hamiltonians for $XY[0,-(\pi /2)]$ (blue), $XY[\pi /2,\pi /2)$ (orange), and $XY(0,\pi )$ (pink) ($\mathrm{\Delta }t=4\mathrm{ns}$). The shaded band around each Pauli term represents the confidence interval, obtained from the average Pauli amplitude when no coupler pulse was applied. Single-qubit and additional two-qubit Pauli terms have no appreciable signal (see Appendix pp7) (d) The reconstruction infidelity (averaged over initial states) computed with respect to tomography after the pulse is small for all three pulses. Preconditioning the Hamiltonian reconstruction algorithm with an $IZ$ and $ZI$ term (magnitude derived from QPT) reduces this error further (right, shaded bars), particularly for $XY(0,\pi )$. (e) Dynamical gate infidelity during the three pulses showing the accuracy of the reconstructed dynamics with respect to the assumed pulse. Inset shows dynamical gate fidelity for pulse (iii) in Fig. 2 and reveals large deviation from an “ideal” control Hamiltonian despite high final gate fidelity.

Figure 5

Experimental setup. Details and specification of larger electronics are given in Appendix pp1.

Figure 6

Average reconstruction fidelity versus critical frequency of low-pass Butterworth filter applied to the data. Dashed line is provided as a guide for the eye. The reconstruction fidelity, averaged over initial states, $⟨{\mathcal{F}}_r⟩$ falls when the time series voltage data is filtered too strongly and distorts the dynamics.

Figure 7

Full set of reconstructed terms for two single-qubit $\pi$ pulses around the $x$ axis. Solid (dashed) lines represent reconstructed (estimated) amplitudes, with time resolution $\mathrm{\Delta }t=4\mathrm{ns}$. The shaded band indicates an uncertainty, calculated from the “No pulse” data. ${\mathrm{\Omega }}_{IZ}(t),{\mathrm{\Omega }}_{ZI}(t),\text{ and }{\mathrm{\Omega }}_{ZZ}(t)$ are not solved for, due to the constraint of the first-order update. No significant crosstalk is observed from this reconstruction.

Figure 8

Full set of reconstructed terms for the entangling two-qubit Hamiltonians $XY(\beta ,\theta )$ explored in this work. Solid (dashed) lines represent reconstructed (estimated) amplitudes, with time resolution $\mathrm{\Delta }t=4\mathrm{ns}$. ${\mathrm{\Omega }}_{IZ}(t),{\mathrm{\Omega }}_{ZI}(t),\text{ and }{\mathrm{\Omega }}_{ZZ}(t)$ are not solved for, due to the constraint of the first-order update, but ${\mathrm{\Omega }}_{IZ}(t)$ and ${\mathrm{\Omega }}_{ZI}(t)$ are preconditioned as described in the main text. Negligible amplitude appears in the Pauli terms outside the center four, so those are presented in the main text.