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10-qubit entanglement and parallel logic operations with a superconducting circuit
Chao Song, Kai Xu, Wuxin Liu, Chuiping Yang, Shi-Biao Zheng, Hui Deng, Qiwei Xie, Keqiang Huang, Qiujiang Guo, Libo Zhang, Pengfei Zhang, Da Xu, Dongning Zheng, Xiaobo Zhu, H. Wang, Y. -A. Chen, C. -Y. Lu, Siyuan Han, J. -W. Pan
TL;DR
The paper addresses the challenge of scaling solid-state multiqubit entanglement, where coherence and sequential gates constrained prior demonstrations. It uses resonator-mediated interactions in a superconducting circuit to produce and tomographically characterize GHZ states, reporting a 10-qubit state fidelity of 0.668 ± 0.025.
Problem
Noise control challenged multiqubit entanglement, and the largest prior solid-state demonstration had five qubits, with further scaling constrained by coherence and sequential-gate methods.
Source: p. 1Method
A superconducting processor uses programmable bus-resonator-mediated couplings to generate GHZ states through a collective interaction and characterize them with quantum state tomography.
Source: p. 1Results
The 10-qubit GHZ state has a measured fidelity of 0.668 ± 0.025, confirming genuine ten-partite entanglement with 6.7 standard deviations.
Source: pp. 1, 3Takeaways & Limitations
The experiment demonstrates the largest multiqubit entanglement then created in solid-state systems and verifies it by quantum state tomography.
Source: p. 4Single-qubit gate performance may slightly degrade after a large square frequency-tuning pulse because of its finite rise time.
Source: XY control · p. 7
Abstract
from arXiv · showhide
Here we report on the production and tomography of genuinely entangled Greenberger-Horne-Zeilinger states with up to 10 qubits connecting to a bus resonator in a superconducting circuit, where the resonator-mediated qubit-qubit interactions are used to controllably entangle multiple qubits and to operate on different pairs of qubits in parallel. The resulting 10-qubit density matrix is unambiguously probed, with a fidelity of $0.668 \pm 0.025$. Our results demonstrate the largest entanglement created so far in solid-state architectures, and pave the way to large-scale quantum computation.
Device
The device is a 10-Xmon superconducting circuit interconnected by a central bus resonator. Each qubit can be tuned between an idle frequency and a shared interaction frequency for entangling operations.
Device
- The sample has 10 Xmon qubits interconnected by a bus resonator, a half-wavelength coplanar-waveguide resonator fixed at approximately 5.795 GHz, measured with coupling qubits in |0⟩ at idle frequencies.Source: Device · p. 5
- Individual flux lines dynamically tune qubit frequencies, while microwave drives control each qubit’s |0⟩↔|1⟩ transition.Q2, Q4, and Q6 share other qubits’ microwave lines in this experiment.Source: Device · p. 5
- Qubits are initialized and measured at idle frequencies 4.96–5.66 GHz, then biased to an interaction frequency for a specified period before returning to idle.The idle frequencies are 140–840 MHz below the bus-resonator frequency.Source: Device · p. 5
XY control
XY controls deliver rotations and readout tones across the 10-qubit device, with randomized benchmarking showing high single-qubit gate fidelities. Gate performance can slightly degrade after frequency-tuning pulses, and the experiment allows time for recovery before applying rotations.
XY control
- Seven independent xy signal channels provide 10 tones for controlling 10 qubits, including two tones on each of three channels.The tones are generated as sidebands mixed with a continuous microwave carrier at 5.324 GHz.Source: XY control · p. 5
- Each qubit has a readout resonator, and a 1 µs readout pulse carries 10 tones to target the 10 resonators.The table also defines readout fidelity as the probability of correctly identifying a qubit prepared in |0⟩ or |1⟩.Source: XY control · p. 6
- X/2 gate fidelities are at least 0.998 on all 10 qubits, and at least 0.993 during simultaneous randomized benchmarking on eight selected qubits.Simultaneous benchmarking on neighboring Q8 and Q9 found fidelities of at least 0.997 with shorter sequences.Source: XY control · p. 7
- Optimized single-qubit gates on Q5 have fidelities above 0.999 for X, Y, ±X/2, and ±Y/2 rotations.The optimization shortens gate time and slightly detunes the xy pulse to reduce phase error.Source: XY control · p. 7
- After frequency-tuning square pulses, single-qubit gate performance may slightly degrade because of finite pulse rise time; the experiment waits 10 ns before applying gates.The rise time is on the order of a few to a few tens of nanoseconds for an ideally sharp step edge.Source: XY control · p. 7
- After π/2 rotations transform the seven-qubit state into the |0⟩ and |1⟩ basis, its measured fidelity is 0.771 ± 0.022.The comparison with the pre-rotation fidelity estimates average post-pulse π/2 rotation fidelity above 0.995; simulation suggests above 0.993.Source: XY control · p. 7
Z control
Ten independent z channels allow simultaneous qubit-frequency tuning. The team calibrates and compensates for finite pulse rise times and measures z-crosstalk, which reaches about 8% between neighboring qubits.
Z control
- Ten independent z channels provide the capacity to tune all 10 qubit frequencies simultaneously.Finite square-pulse rise times are corrected by deconvolving measured step-edge responses from room-temperature wiring.Source: Z control · p. 7
- The calibrated z-crosstalk matrix maps applied biases to actual qubit biases, with crosstalk reaching about 8% between neighboring qubits.Cryogenic wiring imperfections are partially compensated using qubit transition-frequency response to a step-edge bias.Source: Z control · p. 7
- The authors note that z-crosstalk may not contribute to GHZ-state errors because they iteratively fine-tune each qubit’s z bias over a small range.Source: Z control · p. 7
Qubit readout
The study characterizes multiplexed qubit readout and calibrates residual nearest-neighbor coupling. Including measured XX-type crosstalk in GHZ-state optimization lowers predicted fidelities for N > 6 but raises the N = 10 fidelity.
Qubit readout
- Multiplexed readout tones encode qubit states in I-Q signals, with a JPA amplifying the signals before simultaneous demodulation.The readout pulse is 1 µs long, and the resonator response yields I-Q values used to distinguish |0⟩ from |1⟩.Source: Qubit readout · pp. 7, 9
- For Q3 and Q4, choosing readout tones and powers limits crosstalk, though Q3 readout fidelity decreases slightly when both are measured.With readout correction, product-state preparation and one- and two-qubit tomography yield reconstructed density-matrix fidelities around or above 0.99.Source: Qubit readout · p. 10, Figure S3 · p. 10
- A direct Q8–Q9 XX-type coupling of approximately 2π × 2.1 MHz is needed alongside resonator-mediated superexchange to explain the measured interaction.The inferred net interaction agrees with numerical simulation including the extra coupling; the table lists estimated nearest-neighbor couplings.Source: Qubit readout · pp. 10–11, Table S2 · p. 11
- Including estimated XX-type nearest-neighbor couplings during numerical optimization of GHZ-state parameters lowers fidelities for N > 6 but raises fidelity for N = 10.Numerical results agree reasonably well with data in the first few hundred nanoseconds; GHZ sequence times are less than 150 ns.Source: Qubit readout · p. 11
MULTIQUBIT GHZ PHASE CALIBRATION
The pairwise calibration adjusts microwave phases to account for dynamical phases accumulated during frequency changes. With Q3 as reference, the calibrated pairs show equal probabilities for the four computational states under the stated interaction condition.
MULTIQUBIT GHZ PHASE CALIBRATION
- Calibrating each qubit’s drive phase to the rotating frame at ωI accounts for extra dynamical phases accumulated during frequency adjustment.The calibration seeks zero relative phase after the two qubits are placed on resonance.Source: MULTIQUBIT GHZ PHASE CALIBRATION · p. 11
- For negative λ, the two-qubit state has equal probabilities across the four computational states when ϕ = 0.Source: MULTIQUBIT GHZ PHASE CALIBRATION · p. 11
- The experiment uses Q3 as the reference and adjusts the other qubit’s microwave phase pairwise, with calibrated phase-dependent occupation probabilities shown in Fig. S5.Source: MULTIQUBIT GHZ PHASE CALIBRATION · pp. 11–12, Figure S5 · p. 12
Effect of reduced sample size
Full N-qubit tomography requires many operations and samples, making measurement lengthy. Tests for N = 6 to 9 report that about 3 × 2^N samples can yield fidelity close to that from a sufficiently large sample.
Effect of reduced sample size
- At N = 10, full tomography uses 3^N operations and 2^N occupation probabilities per operation; 3000 samples make the uninterrupted measurement take about 40 hours.The system must be monitored during measurement to ensure reasonably stable performance.Source: Effect of reduced sample size · p. 12
- For N = 6 to 9, sample sizes around 3 × 2^N produce reconstructed density matrices with fidelity very close to those from sufficiently large samples.Source: Effect of reduced sample size · p. 12, Figure S6 · p. 12
Reducing the computation complexity
The tomography data are stacked into a linear system for the vectorized density matrix. Sparse storage and solving the normal equations reduce the computation burden as the number of qubits grows.
Reducing the computation complexity
- Stacking the tomographic operation matrices and measured probabilities gives the linear system Ũρ̃ = P̃ for solving the vectorized density matrix.The operation matrices and probability vectors are combined across all 3^N tomographic operations.Source: Reducing the computation complexity · pp. 12–13
- Although Ũ becomes difficult to load as N approaches 10, its small fraction of nonzero elements permits sparse storage and a pseudo-inverse approach.The normal-equation matrix Ũ†Ũ is smaller and sparser than Ũ, reducing the complexity of solving the equations.Source: Reducing the computation complexity · p. 13
- The method estimates the matrix-inversion cost for general full-rank matrices and compares it with the sparse structure of Ũ†Ũ.The supplied passages describe the comparison but do not provide a complete complexity expression.Source: Reducing the computation complexity · p. 13
10-qubit ρ in the |0⟩and |1⟩basis
The 10-qubit GHZ density matrix is shown in the |0⟩ and |1⟩ basis, where every reported matrix element has amplitude no higher than 0.003. The matrix has four major elements in the |±⟩ basis.
10-qubit ρ in the |0⟩and |1⟩basis
- In the |0⟩ and |1⟩ basis, all 10-qubit GHZ density-matrix elements are no higher than 0.003 in amplitude.The partial matrix elements are shown in Fig. S7.Source: 10-qubit ρ in the |0⟩and |1⟩basis · p. 13, Figure S7 · p. 14
- The density matrix has four major elements in the |±⟩ basis, unlike the basis used for the measurement.Source: 10-qubit ρ in the |0⟩and |1⟩basis · p. 13
- Fig. S7 presents partial matrix elements for the 10-qubit GHZ density matrix in the |0⟩ and |1⟩ basis.Source: 10-qubit ρ in the |0⟩and |1⟩basis · p. 13, Figure S7 · p. 14