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Digitized adiabatic quantum computing with a superconducting circuit
R. Barends, A. Shabani, L. Lamata, J. Kelly, A. Mezzacapo, U. Las Heras, R. Babbush, A. G. Fowler, B. Campbell, Yu Chen, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, E. Lucero, A. Megrant, J. Y. Mutus, M. Neeley, C. Neill, P. J. J. O'Malley, C. Quintana, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, T. C. White, E. Solano, H. Neven, John M. Martinis
TL;DR
Analog quantum computing faces precision and coherence challenges, while digitized adiabatic quantum computing is proposed for execution on error-corrected digital devices. The approach discretizes continuous evolution into equal time steps and compares experimental, ideal, and target-state fidelities, while residual energy decreases as a power law with simulated time for large Ising chains.
Problem
Analog quantum annealers require high-precision interaction tuning, while decoherence can limit analog quantum computers without established fault-tolerant error correction.
Method
The paper promotes digitized adiabatic quantum computing by simulating adiabatic evolution with single- and two-qubit gates using equal-size time discretization and first-order Trotterization.
Results
For large Ising spin chains, residual energy follows a power law in simulated time, with visible exponents η > 0.5; the experiment also compares experimental and ideal evolution fidelities with target-state results.
Takeaways & Limitations
Digitized adiabatic quantum computing is presented as an approach without a fundamental limit on the precision and coherence-related features considered for analog systems, subject to qubit cost.
Takeaways & Limitations
The experiment used first-order Trotterization because of experimental limitations, and substantial overhead in L, k, and γ^-1 makes it unlikely to be practical.
Abstract
from arXiv · showhide
A major challenge in quantum computing is to solve general problems with limited physical hardware. Here, we implement digitized adiabatic quantum computing, combining the generality of the adiabatic algorithm with the universality of the digital approach, using a superconducting circuit with nine qubits. We probe the adiabatic evolutions, and quantify the success of the algorithm for random spin problems. We find that the system can approximate the solutions to both frustrated Ising problems and problems with more complex interactions, with a performance that is comparable. The presented approach is compatible with small-scale systems as well as future error-corrected quantum computers.
Supplementary Information for “Digitized adiabatic quantum computing with a superconducting
The supplementary information identifies the document as supplementary material for “Digitized adiabatic quantum computing with a superconducting.”
- The listed author affiliation is IBM T. J. Watson Research Center in Yorktown Heights.
- The passage specifies a present address rather than describing the paper’s methods or findings.
I. WHY DIGITIZED ADIABATIC QUANTUM COMPUTING?
Digitized AQC is proposed to retain AQC’s general-purpose optimization capability while addressing hardware limitations of analog quantum annealers through programmable gate-based interactions.
- Digitized AQC is promoted as a viable algorithm for execution on an error-corrected digital quantum device.
- AQC is a general-purpose optimization algorithm that can, in principle, map any optimization problem to an adiabatic computation.
- Analog annealers face constraints involving graph connectivity, k-body interactions, arbitrary interactions, precision, and coherence.These constraints can limit programmable problem Hamiltonians and device performance.
- A digital approach simulates AQC with single- and two-qubit gates and has no fundamental limit on achieving the listed annealer features, at a cost in required qubits.
II. METHODS OF DIGITIZATION AND DISCUSSION OF SCALING
The paper digitizes time-dependent adiabatic evolutions with first-order Lie-Trotter-Suzuki steps and derives gate-complexity scaling by combining digitization and adiabatic runtime bounds.
- The experiments use first-order Lie-Trotter-Suzuki digitization for adiabatic quantum computing.
- A time-dependent Hamiltonian is decomposed into L local k-local terms with time-dependent scalar coefficients, then discretized into M equal steps of size δt = T/M.
- O(MLk) gates implement the digitized evolution when arbitrary rotations are available and each k-local Hamiltonian uses at most O(k) gates.
- The digitization error is controlled by choosing M according to the total evolution time, coefficient bound, number of Hamiltonian terms, and target error ϵ.
- The adiabatic runtime is governed by the minimum spectral gap γ during the evolution, linking the gap to the required gate count.
- First-order Trotterization was chosen for experimental reasons, while its overhead in L, k, and γ^-1 makes it unlikely to be practical; higher-order schemes may improve scaling.
III. RESIDUAL ENERGY SCALING
Residual energy provides a practical measure of how closely digitized adiabatic evolution approaches the ground state. The experiment shows a short-time plateau followed by power-law decay, with η > 0.5 visible in the measured regime.
- Residual energy measures the energy above the problem Hamiltonian’s ground state, with smaller values indicating better optimization outcomes.
- The Kibble-Zurek picture links faster passage through the phase transition to insufficient spin adjustment and residual kink defects.
- For fast quenches, residual energy remains nearly time-independent because the system has insufficient time to adjust to lower-energy states.
- The expected kink density and residual energy scale approximately as T^-1/2 through the longest defect-free chain N* ∝ T^1/2.
- η > 0.5: the experiment visibly transitions from a short-time plateau to power-law residual-energy decay.
IV. PAIRWISE INTERACTION IN A NINE QUBIT SYSTEM
The nine-qubit chain uses frequency control to alternate between minimized interactions and selectively activated nearest-neighbour interactions. This enables pairwise gates while decoupling other qubits.
- Alternating qubit frequencies minimize parasitic interactions during idling in the nine-qubit chain.
- Adjacent qubits are typically detuned by 1 GHz, while next-nearest qubits are detuned by 0.1 GHz.
- To entangle a pair, one qubit is moved to a higher frequency while its neighbour follows an adiabatic trajectory producing a conditional phase shift.
- Qubits adjacent to the interacting pair are decoupled with pulses to reduce unwanted participation.
V. CONSTRUCTING INTERACTION
The experiment constructs tunable two-qubit interactions with a calibrated CZφ gate and single-qubit phase corrections. The implementation covers σzσz couplings while respecting a finite controllable-phase range.
- The CZφ gate tunes conditional phase by adiabatically sweeping one qubit near the |02⟩–|11⟩ interaction.
- The interaction layout uses idling qubits, adjacent CZφ gates, and π-pulse-decoupled qubits.
- Careful calibration nulls residual single-qubit phases to within 0.05 radians while achieving the desired conditional phase.
- The CZφ phase is limited to approximately 0.5–4.5 because lower values suffer interaction complications and higher values cause leakage.
- For |φ| > 2.25, two entangling gates or 2π adjustments extend implementation, with φ = 2Jzz∆t realizing U = exp(−iJzz∆t).
VI. DECOUPLING FROM THE ENVIRONMENT AND PARASITIC INTERACTIONS
The pulse sequences combine Trotterized evolution with decoupling operations that suppress dephasing and parasitic interactions. Local fields and both σzσz and σxσx couplings are compiled into quantum logic gates.
- Correlated dephasing and parasitic interactions with other qubits are important environmental error sources.
- Decoupling π pulses surround σzσz and σxσx interactions, interrupt idling exposure, and isolate neighbours during controlled-phase trajectories.
- Trotter expansion divides the continuous evolution into small time steps, each implemented as a sequence of quantum logic gates.
- Single-qubit gates implement local fields, while one or two CZφ gates combined with single-qubit gates implement σzσz and σxσx couplings.
- Pulse-sequence diagrams distinguish slow rectangular frequency-detuning pulses from rapidly oscillating microwave pulses.
3. Second Trotter step
The second Trotter-step implementation combines single-qubit rotations, tunable entangling gates, and decoupling pulses within the digitized evolution. The supplementary materials document the sequence timing, configurations, and gate counts.
- Sequence accounting: Table S1 counts gates across initialization and all Trotter steps, treating durations of 10 ns or longer as idles.
- Interaction implementation: The first non-stoquastic Trotter step includes both σzσz and σxσx interactions, with other qubits waiting when two entangling gates are required.
- Interaction implementation: σxσx interactions use π/2 pulses for basis rotation before entangling operations.
- Interaction implementation: In the second Trotter step, the σxσx interaction can use a single CZφ gate as coupling strength and phases increase linearly.
- Timing: Trotter steps 2–5 are shorter than the first step.
- Sequence accounting: The supplementary information provides experiment-wide Trotter-step counts, simulated times, couplings, and field strengths.
IX. DIGITAL EVOLUTION INTO GHZ STATE: IMAGINARY PARTS AND IDEAL ADIABATIC EVOLUTION
At s = 1.0, the supplementary analysis compares real and imaginary components with the ideal adiabatic evolution and target state. The ideal evolution is closer to the target than the experimental data are to the ideal evolution.
- At s = 1.0, the real and imaginary parts of the evolution are shown alongside the ideal adiabatic evolution and target state.
- The ideal adiabatic evolution has fidelity 0.92 with respect to the target state.
- The experimental data have fidelity 0.60 with respect to the ideal adiabatic evolution.
X. KINK LIKELIHOOD FOR TWO TO NINE QUBIT CONFIGURATIONS
Kink likelihood generally shifts toward kink-free configurations as simulation time increases, although larger systems show a later resurgence of kinks. Residual-energy deviations identify digitization error near |J|T = 3 for nine qubits.
- Kink likelihood: For two qubits, the only one-kink configurations are |01⟩ and |10⟩, while initially zero and one kink are equally likely.
- Kink likelihood: As simulation time increases, kink likelihood decreases and the likelihood of no kinks increases across the systems.
- Kink likelihood: For seven- to nine-qubit systems, kink likelihood rises again around |J|T = 2.
- Residual energy: Near |J|T = 3, the nine-qubit residual-energy difference increases because of digitization error, while experiment and ideal digital evolution differ by at most 2|J|.
XI. SPIN PARITY CORRELATION
The five-qubit antiferromagnetic experiment exhibits parity correlations whose signs depend on distance and agrees with theory. The supplementary calibration procedure separately characterizes microwave rotations and tunable CZφ phases.
- Spin parity correlation: For the five-qubit antiferromagnetic experiment, parity correlations are negative at odd distances and positive otherwise.
- Spin parity correlation: Measured parity correlations follow the theory predictions for either direction.
- Calibration: Variable microwave rotations are calibrated from measurement probabilities as a function of amplitude.
- Calibration: The tunable CZφ phase is calibrated by preparing the static qubit on the equator, varying the other trajectory amplitude, and using tomography.
XIII. COMPARISON OF FIDELITIES BETWEEN EXPERIMENT, IDEAL DIGITAL EVOLUTION, IDEAL CONTINUOUS EVOLUTION, AND TARGET STATE RESULTS, AS WELL AS UNIFORM RANDOM VALUES
This section compares experimental fidelities with ideal digital, ideal continuous, target-state, and uniformly random baselines, using overview figures, tables, and example-instance results.
- Fidelities are compared across experiment, ideal digital evolution, ideal continuous evolution, and the target state.The comparisons are reported as fidelities, with uniformly generated random probabilities included as a baseline sanity check.
- Example instances from the main article are summarized through fidelities and success measures in Table S10.The table covers experimental data, ideal digital evolution, ideal continuous evolution, and target-state comparisons.
- Tables S2–S8 provide simulation parameters and problem instances for random, stoquastic, and non-stoquastic experiments.The listed instances include three-, six-, seven-, and nine-qubit configurations.
- Supplementary figures examine kink likelihood, residual-energy differences, spin-parity correlations, gate calibrations, pulse sequences, and GHZ-state evolution.These diagnostics span ferromagnetic chains, antiferromagnetic-state evolution, control calibration, and representative stoquastic or non-stoquastic experiments.
- The fidelity overview aggregates normalized histograms and success measures for both stoquastic and non-stoquastic problems.Experimental results are compared with ideal digital and ideal continuous evolutions.