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Implementing the Quantum von Neumann Architecture with Superconducting Circuits
Matteo Mariantoni, H. Wang, T. Yamamoto, M. Neeley, Radoslaw C. Bialczak, Y. Chen, M. Lenander, Erik Lucero, A. D. O'Connell, D. Sank, M. Weides, J. Wenner, Y. Yin, J. Zhao, A. N. Korotkov, A. N. Cleland, John M. Martinis
TL;DR
The paper develops a programmable quantum von Neumann architecture using coupled superconducting qubits, a quantum bus, and quantum memories. It derives, tunes, and measures CZ-φ gates, including phases 0.01, π/2, and π, while characterizing calibration and measurement constraints.
Problem
The paper addresses how a quantum von Neumann architecture can implement programmable operations involving a bus-mediated interaction between superconducting qubits.
Method
The authors derive CZ-φ gate dynamics from effective Hamiltonians, use qutrit and bus-resonator states, and calibrate phases with Ramsey experiments.
Results
Ramsey-fringe phase differences measure CZ-0.01, CZ-π/2, and CZ-π gate phases, while φ = π occurs at zero Q1-B detuning and 0 ≲ φ < π for positive detuning.
Takeaways & Limitations
The calibration procedure cancels unwanted dynamic phases and supports selecting a desired CZ-φ phase through a localized search near theoretically estimated parameters.
Takeaways & Limitations
During the XOR gate, resonator B can be controlled only before or after the operation because qubit Q2 is actively used during the gate.
Abstract
from arXiv · showhide
The von Neumann architecture for a classical computer comprises a central processing unit and a memory holding instructions and data. We demonstrate a quantum central processing unit that exchanges data with a quantum random-access memory integrated on a chip, with instructions stored on a classical computer. We test our quantum machine by executing codes that involve seven quantum elements: Two superconducting qubits coupled through a quantum bus, two quantum memories, and two zeroing registers. Two vital algorithms for quantum computing are demonstrated, the quantum Fourier transform, with 66% process fidelity, and the three-qubit Toffoli OR phase gate, with 98% phase fidelity. Our results, in combination especially with longer qubit coherence, illustrate a potentially viable approach to factoring numbers and implementing simple quantum error correction codes.
Statistical errors
The analysis models statistical uncertainty in qubit tomography through binomial measurement errors and phase fluctuations, then propagates these errors to density matrices and their metrics. Confidence intervals are estimated by simulating ensembles when independent measurements are insufficient and directly from repeated measurements when available.
- Confidence intervals: Confidence intervals for density-matrix elements and metrics are obtained from simulated or independently measured ensembles, depending on the available data.The metrics include fidelity, negativity, concurrence, and entanglement of formation.
- Error sources: Binomial-type errors arise from repeated two-outcome qubit measurements, while phase errors arise mainly from jitter in room-temperature cables and electronics.The measurement process is modeled with statistically independent Bernoulli trials, and phase fluctuations are quantified experimentally.
- Tomography protocols: Two-qubit tomography uses 9 operation settings and 36 probabilities, whereas octomo uses 36 settings and 144 probabilities.Each setting produces four joint probabilities; the effective event counts are 9 for tomo and 36 for octomo.
- Simulation procedure: The analysis corrects estimated probabilities for measurement errors, computes their binomial standard deviations, and generates M = 1000 normally distributed perturbation samples.Each simulated sample produces a different QST experiment, allowing density-matrix and metric distributions to be calculated.
- Simulation results: 0.95 confidence interval: the real part of ⟨eg|ρπ|ge⟩ has mean 0.338 and ±2σb = ±0.005 in the octomo simulation.The resulting distribution is approximately Gaussian.
Definition of the qubit reference frame
The qubit reference frame is set by microwave driving at the idle-point transition frequency, while z-pulses create detuning and dynamic z-axis phases that experiments compensate.
- Reference frame: A z-pulse tunes the qubit transition frequency and changes its detuning from the reference clock rate.The detuning is the difference between f_Q(z) and the reference clock rate f_Q^0.
- Driven dynamics: Microwave pulses prepare excited or superposition states through a driven qubit Hamiltonian.The driving Hamiltonian depends on the time-varying amplitude, driving frequency, Pauli operator, time, and phase delay.
- Reference frame: The qubit reference frame uses a fixed driving frequency equal to the transition frequency at the idle point.The idle point is defined by z = 0, with reference clock rate f_Q^0.
- Phase compensation: Leaving the reference frame produces dynamic rotations about the z-axis, which are compensated during experiments.The total driven-system Hamiltonian combines the qubit and driving Hamiltonians.
Programming the quantum von Neumann architecture
The architecture uses calibrated phase-sensitive control and tomography to program interactions between qubits, resonators, and memories, including the tunable CZ-φ gate.
- Phase control: Dynamic phases arise when qubits leave their reference frames and are calibrated or compensated during pulse sequences.The measured phase differences in density matrices agree with calculated phases, while later experiments use compensation pulses directly.
- Characterization: Density matrices and process matrices are reconstructed with confidence intervals for their real and imaginary components.Table S2 reports density-matrix elements for φ = 0.28, π/2, and π.
- CZ-φ gate: The CZ-φ interaction tunes the acquired phase through the detuning between |f0⟩ and |e1⟩.Resonance produces φ = π, while positive detuning yields phases from approximately 0 to below π.
The quantum Fourier transform
The quantum Fourier transform relies on a tunable CZ-φ gate mediated by a bus resonator, with pulse calibration and compensation enabling phase control across the algorithm.
- CZ-φ gate: The CZ-φ gate uses Q1 as a qutrit target, Q2 as a control, and bus resonator B to mediate their interaction.The active third state |f⟩ of Q1 participates in the gate dynamics.
- CZ-φ gate: The gate phase is acquired during a full 2π interaction between |f0⟩ and |e1⟩.The phase is π at zero detuning and spans approximately 0 ≤ φ < π for positive detuning.
- Reference frames: Independent qubit reference frames can be used without a special phase relationship between their frequencies.The gate’s phase behavior is insensitive to the relative phases of Q1 and Q2 when brought into resonance through B.
- Calibration: The CZ-φ phase is measured from the difference between Ramsey fringes with and without pulses on Q2.This procedure measures phases for CZ-0.01, CZ-π/2, and CZ-π gates.
- Calibration: The compensation-pulse calibration is cross-checked by comparing Ramsey-fringe extrema between the two sequences.For φ = π, the second sequence reaches a minimum at the compensation value where the first reaches a maximum.
- Calibration: Compensation pulses cancel unwanted dynamic phases accumulated during the gate sequences.Choosing the maximum of Q2’s excitation probability cancels its dynamic phase from the two iSWAP operations.
XOR gate and M gate tuneup
The XOR and M gates are tuned using Ramsey-type sequences that measure and compensate dynamic phases on the control qubits and target resonator. The XOR tuneup requires an additional resonator-focused sequence because the bus acts as the third qubit.
- Overview: The tuneup procedure covers both control-qubit and target-resonator phases required for the three-qubit XOR and M gates.The supporting procedures describe pulse sequences and Ramsey fringes for calibrating the gates’ phase behavior.
- XOR gate: Ramsey experiments determine and compensate dynamic phases acquired during the XOR gate.The first two sequences act on control qubits Q1 and Q2, using compensation pulses selected where the measured excited-state probability reaches a maximum.
- XOR gate: The third XOR sequence calibrates the target resonator B indirectly through qubit Q2.Q2 writes a superposition into B through an iSWAP, and a second iSWAP returns the state for Ramsey readout.
- XOR gate: τdel ≃3.46 ns maximizes the Ramsey fringe used to compensate B’s dynamic phase in the XOR gate.The delay is chosen at a fringe maximum after the state in B acquires phase from detuning relative to Q2’s reference clock.
M gate tuneup
The M gate is calibrated through six Ramsey-based sequences addressing control-qubit phases, shelving phases, and target-resonator phases. The procedure compensates these phases with pulse amplitudes, delays, and detuning choices.
- M gate tuneup: Six tuneup sequences calibrate the M gate’s control-qubit, shelving, and resonator dynamic phases.The sequence set extends the XOR tuneup to account for the split 1/2 CZ-π gates and shelving dynamics.
- Control-qubit calibration: zcmp ≃−0.119 and zcmp ≃−0.077 maximize the Ramsey probabilities for Q1 and Q2, respectively.The first and second sequences calibrate compensation amplitudes for the two control qubits.
- Shelving-phase calibration: The third sequence compensates Q1’s shelving phase by varying τdel1 between the two 1/2 CZ-π gates.Ramsey fringes differ according to whether the target resonator is in |0⟩ or |1⟩, revealing the shelving-dependent phase.
- Resonator calibration: τdel2 ≃2.77 ns is one delay choice that maximizes Q2’s Ramsey probability when calibrating the target resonator.The delay differs slightly from the XOR value because the experiments were separated by several hours and qubit frequencies drifted.
- Resonator calibration: zdet ≃−0.012 compensates the dynamic phase acquired by B during shelving, followed by recalibration of Q1 with zcmp ≃−0.125.The final sequence repeats the first M-gate tuneup after applying the resonator detuning.
M gate pulse sequence
The M-gate pulse sequence loads a superposition into the target resonator, applies the two-control gate, compensates dynamic phases, zeros Q2, and reads out the resonator through Q2.
- State preparation: Q2 is prepared in a superposition and transfers its state into B through an iSWAP.The iSWAP zeros Q2 so it can subsequently serve as a control qubit.
- Gate operation: The gate applies two 1/2 CZ-π operations between Q1 and B with a CZ-π operation between Q2 and B.The shelving interval also includes the delay and detune used to compensate dynamic phases on Q1 and B.
- Phase compensation and readout: Compensation pulses and delay are followed by a zeroing gate on Q2 before Ramsey readout of B.A final Rπ/2^y rotation on Q2 and measurement complete the indirect Ramsey experiment on the resonator.
Quantum phase tomography
Quantum phase tomography reconstructs the independent phases of three-qubit controlled-phase gates from Ramsey-measured phase differences. A phase-gate cube and a rank-seven linear system organize this reconstruction for the XOR and M gates.
- Phase representation: Three-qubit controlled-phase gates have seven physically independent phases after removing one global phase.The eight diagonal phases are represented relative to the phase of |gg0⟩.
- Tomography construction: Ramsey experiments measure twelve phase differences, which are converted into seven gate phases through a phase-gate cube.The cube’s eight vertices represent diagonal gate elements, while its twelve edges represent measured phase differences.
- Linear reconstruction: The transformation matrix Tϕτ has dimensions (12, 8) and rank 7.Rows correspond to cube segments and columns to vertices, matching the seven independent phases after the global phase is removed.
- Linear reconstruction: Because Tϕτ is not invertible, the measured phase differences are fitted using an overconstrained least-squares system.The fitted solution yields the reconstructed vector τ for the gate phases.
- Experimental reconstruction: Figure S12 uses twelve Ramsey fringes for each gate and propagates fit confidence intervals into the reconstructed phase uncertainties.Panels A and B show probability versus Ramsey phase; panels C and D show the resulting phase differences for XOR and M.