Analog-Digital Quantum Computing with Quantum Annealing Processors

Abstract

Quantum annealing processors typically control qubits in unison, attenuating quantum fluctuations uniformly until the applied system Hamiltonian is diagonal in the computational basis. This simplifies control requirements, allowing annealing QPUs to scale to much larger sizes than gate-based systems, but constraining the class of available operations. Here we expand the class by performing analog-digital quantum computing in a highly-multiplexed, superconducting quantum annealing processor. This involves evolution under a fixed many-body Hamiltonian that, in the weak-coupling regime, is well-described by an effective XY model, together with arbitrary-basis initialization and measurement via auxiliary qubits. Operationally, this is equivalent to implementing single-qubit gates at the beginning and end of an analog quantum evolution. We demonstrate this capability with several foundational applications: single-qubit and two-qubit coherent oscillations with varying initialization and measurement bases, a multi-qubit quantum walk with fermionic dispersion in line with theory, and Anderson localization in a disordered chain. These experiments open the door to a wide range of new possibilities in quantum computation and simulation, greatly expanding the applications of commercially available quantum annealing processors.

Figures (15)

• Figure 1: Analog-digital protocol in a quantum annealer.a, The time-dependent quantum annealing (QA) Hamiltonian $H(t)$\ref{['eq:H_anneal']} is governed by transverse and Ising energy scales $A(s)$ and $B(s)$ and a control parameter $s(t)$. b, In standard QA, $s(t)$ increases linearly from $0$ to $1$, interpolating $H(t)$ from a transverse-field-dominated Hamiltonian to one dominated by the problem Hamiltonian. c, Anneal schedules $s^\alpha(t)$ are controlled independently on several annealing lines $\alpha$, not necessarily in unison. Our analog-digital setup reserves one or more lines for source and detector qubits, which respectively prepare and measure the states of the remaining "target" qubits. d, Free evolution of the target qubits under the action of a time-independent many-body analog Hamiltonian $H_\text{target}(s^*)$ is initiated by a quench of the source qubits to $s{=}0$ and terminated by a quench of the detector qubits to $s{=}1$; Operationally, these processes implement individually programmable single-qubit gates before and after the evolution under $H(s^*)$, enabling very fast (on the time scale of target qubit interactions) initialization and readout.
• Figure 2: Single-qubit Larmor precession with arbitrary-basis initialization and readout.a, Illustration of the protocol: Initial state is excited with an initial rotation gate (orange), then allowed to precess freely as it relaxes back to $\ket 0$ (blue). b, Measured magnetization for a single qubit for varying evolution time $t$ under the Hamiltonian $H_0$. Flux bias is applied to the detector to tilt the measurement basis (top row), interpolating from the $\sigma^x$ basis ($\theta_d=\tfrac{\pi}{2}$) to the $\sigma^z$ basis ($\theta_d\in\{0,\pi\}$) and correspondingly revealing dephasing-dominated and relaxation-dominated dynamics. Applying flux bias to the source qubit (bottom row) rotates the initial excitation from $\ket +$ to $\ket 1$. Short-time data ($t < 2ns$, omitted) reflect overlap between the initialization and detection processes. c, Polar angles $\theta_d$, $\theta_s$ and azimuthal angles $\varphi_d$, $\varphi_s$ extracted from single-qubit Larmor precession fits to Eq. \ref{['eq:bloch_angles']}. Error bars indicate the 95% confidence interval for the median angle across 124 three-qubit source-target-detector systems.
• Figure 3: Two-qubit spin exchange in multiple measurement bases. We measure a two-qubit system with target qubits $q_1$ and $q_2$ at $\Delta=1.0GHz$. In the first two columns, the system is prepared in the product state $\ket{+0}$; in the last two columns, it is prepared in $\ket{10}$. The two-qubit exchange coupling is set to $\mathcal{J}=0$ (left), $\mathcal{J}=0.30GHz$ (middle two columns), or $\mathcal{J}=0.15GHz$ (right). Magnetizations are averaged over 2000 shots. Fitting the relaxation of $\langle\sigma^z\sigma^z\rangle$ and the dephasing of the oscillations in $\langle\sigma^z_1\rangle-\langle\sigma^z_2\rangle$ to an open-quantum-system model SM yields median qubit $T_1 = 30ns$ and $T_\varphi=37ns$ for the third column.
• Figure 4: Excitation propagation in a clean periodic one-dimensional chain. We measure the propagation of two initial states in two bases with $\Delta=2.0GHz$ and $\mathcal{J}=-0.60GHz$. a--b, A single qubit is prepared in the state $\ket +$ in a chain with $L=56$ and evolved under the Hamiltonian in Eqs. \ref{['eq_h0']} and \ref{['eq_hi']}, then measured in the $\sigma^x$ basis. a, Magnetization shows ballistic propagation of information, as well as rays of destructive interference as the light cone reaches the periodic boundary. b, The dispersion relation is given by the peak of the magnetization's Fourier transform, with experimental data in agreement with Eq. \ref{['eq_dispersion']} for evolution time up to $\mathcal{J} t \approx 20$. c--d, A qubit is prepared in the state $\ket 1$ in a chain with $L=124$ and evolved under $H_\text{eff}$, then measured in the $\sigma^z$ basis. c, Again, ballistic propagation is observed throughout the chain. d, The Fourier transform's intensity peak follows Eq. \ref{['eq:photon']}.
• Figure 5: Anderson localization in a disordered chain. Every other qubit is prepared in $\ket 1$ in a periodic chain ($L{=124}$) with disorder $W$ applied through qubit detunings $\delta\Delta_i$ uniformly distributed in $[-\mathcal{J}\tfrac{W}{2}, \mathcal{J}\tfrac{W}{2} ]$, where $\mathcal{J}=0.20GHz$ and the quantizing field is $\Delta=2.0GHz$. In the clean chain ($W=0$), the imbalance oscillates about zero. As disorder is increased, a finite imbalance persists, consistent with localization. Experimental and numerical data are averaged over 20 disorder realizations. Inset: For small $W$, averaged QPU imbalance between $t=15ns$ and $t=20ns$ scales quadratically as $\tilde{\mathcal{I}} = 0.017\,W^{2}$ (gray curve). Error bars indicate uncertainty over time.