Time-resolved tomography of a driven adiabatic quantum simulation

Gian Salis, Nikolaj Moll, Marco Roth, Marc Ganzhorn, and Stefan Filipp

Phys. Rev. A 102, 062611 – Published 17 December 2020

Abstract

A typical goal of a quantum simulation is to find the energy levels and eigenstates of a given Hamiltonian. This can be realized by adiabatically varying the system control parameters to steer an initial eigenstate into the eigenstate of the target Hamiltonian. Such an adiabatic quantum simulation is demonstrated by directly implementing a controllable and smoothly varying Hamiltonian in the rotating frame of two superconducting qubits, including longitudinal and transverse fields and iswap-type two-qubit interactions. The evolution of each eigenstate is tracked using time-resolved state tomography. The energy gaps between instantaneous eigenstates are chosen such that, depending on the energy transition rate, either diabatic or adiabatic passages are observed in the measured energies and correlators. Errors in the obtained energy values induced by finite $T_{1}$ and $T_{2}$ times of the qubits are mitigated by extrapolation to short protocol times.

Received 23 September 2020
Accepted 20 November 2020

DOI:https://doi.org/10.1103/PhysRevA.102.062611

Physics Subject Headings (PhySH)

Adiabatic quantum optimization Quantum information processing Quantum simulation Quantum tomography

Superconducting qubits

Quantum Information, Science & Technology

Authors & Affiliations

Gian Salis^1,*, Nikolaj Moll¹, Marco Roth^1,2, Marc Ganzhorn¹, and Stefan Filipp^1,3

¹IBM Quantum, IBM Research - Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland
²Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany
³Technical University Munich, Department of Physics, 85748 Garching, Germany

^*gsa@zurich.ibm.com

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Vol. 102, Iss. 6 — December 2020

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Figure 1
Adiabatic transition of qubit 2 from the $z$ into the $x$ direction: chirped adiabatic single qubit drive with (a) a frequency offset $f_{2} - f_{i}^{d}$ providing an effective field along the $z$ direction that starts at $z_{2}$ and decreases with time, and (b) amplitude $a_{2}$ of the drive determining the field along $x$ that starts at 0 and linearly increases with time $t$ to $x_{2}$ . (c) Measured evolution of qubit 2 with the two qubits prepared at $t = 0$ in the $| 00 〉$ state when exposed to chirped adiabatic single-qubit drive, as seen in the rotating frame at constant precession frequency $f_{2}$ . (d) Measured expectation values $〈 I X 〉, 〈 I Y 〉$ , and $〈 I Z 〉$ of qubit 2 in the frame at constant precession frequency $f_{2}$ , and (e) in a frame corrected for the accelerating single-qubit drive frequency.
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Figure 2
Calibration of iswap-type gate and dispersive qubit shifts. (a) Measured oscillations of the $| 10 〉$ state occupation after initialization in $| 01 〉$ , for an ac flux modulation amplitude of 0.5. (b) Fitted oscillation frequency of the $| 10 〉$ occupation (symbols) as a function of flux modulation frequency $f_{TC}$ for different modulation amplitudes from 0.2 to 0.9, and corresponding fit of its dependence on $f_{TC}$ (solid lines). (c) Dispersive shifts $f_{i} - f_{i}^{0}$ of individual qubits as induced by driving the iswap-type gate, measured using Ramsey spectroscopy of individual qubits in presence of an ac flux modulation. (d) Strength $j$ of iswap interaction as a function of drive amplitude, as obtained with $f_{TC}$ on resonance [minimum frequency in (b), orange squares]. The solid line is a fit to a third-order polynomial.
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Figure 3
Time-resolved energy levels and Pauli-term expectation values during the adiabatic protocol for $z_{1} / h = 2.5 MHz, z_{2} / h = 1.5 MHz, x_{1} / h = 2.0 MHz, x_{2} / h = 4.1 MHz$ , and $t_{ad} = 10 μ s$ . Because $z_{1} > z_{2}$ but $x_{1} < x_{2}$ , the levels $E_{01}$ and $E_{10}$ cross during the protocol, with a gap that opens with increasing $j$ . Symbols denote measured values, dashed lines exact energy values, and solid lines model calculations taking decoherence and thermalization into account. In (a) and (c), a diabatic passage is observed for $j / h = 0 MHz$ , and in (b) and (d) an adiabatic passage for $j / h = 1.7 MHz$ . In (c) and (d), the expectation values of the relevant Pauli terms are shown for starting in $| 01 〉$ , with a smooth transition in (c) for diabatic passage, and an abrupt sign inversion at the adiabatic passage in (d).
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Figure 4
Influence of length $t_{ad}$ of the adiabatic protocol for $z_{1} / h = 2.5 MHz, z_{2} / h = 1.5 MHz, x_{1} / h = 1.0 MHz, x_{2} / h = 7.3 MHz$ , and $j / h = 1.3 MHz$ . (a) Crossover from diabatic to adiabatic passage. Shown are energy levels $E_{01}$ and $E_{10}$ as a function of $t$ , with a transition from a diabatic passage for $t_{ad} = 5 μ s$ , and an adiabatic passage for $t_{ad} = 30 μ s$ . Symbols are measurements, solid lines calculations. Longer times $t_{ad}$ lead to a stronger decay of the measured expectation values of the Pauli terms, reflected as an overall decrease of the absolute energy values. Individual energy contributions $x_{1} / 2 〈 X I 〉, x_{2} / 2 〈 I X 〉, j / 4 〈 X X 〉$ , and $j / 4 〈 Y Y 〉$ are shown when starting in (b) the ground state $| 00 〉$ and (c) in the highest excited state $| 11 〉$ . The energy contributions decay with longer $t_{ad}$ , which is more pronounced for $| 11 〉$ than for $| 00 〉$ . Open diamonds: measured values; solid lines: fitted second-order polynomials extrapolated to $t_{ad} = 0$ ; filled dots at $t_{ad} = 0$ : calculated energy contributions.
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