Source-linked AI summary
Implementing a strand of a scalable fault-tolerant quantum computing fabric
Jerry M. Chow, Jay M. Gambetta, Easwar Magesan, Srikanth J. Srinivasan, Andrew W. Cross, David W. Abraham, Nicholas A. Masluk, B. R. Johnson, Colm A. Ryan, M. Steffen
TL;DR
The paper addresses fidelity assessment for non-unitary quantum processes, where standard relationships between fidelities are not simple. It develops process and state-tomography methods and reports parity-projection fidelities of 0.90 and 0.91, alongside characterized two-qubit gate and readout performance.
Problem
For non-unitary processes, there is no simple relationship between fidelity measures, motivating appropriate characterization methods.
Method
The study uses a normalized parity-conditioned operation and tomography protocols, including correlation-based state reconstruction and fidelity comparison for quantum operations.
Results
0.90 and 0.91 measurement fidelities are obtained for the odd and even projections, respectively.
Takeaways & Limitations
The characterized parity measurement achieves distinct fidelity metrics for its odd and even projections, while the unconditional map has fidelity 0.968 to complete dephasing in the parity basis.
Takeaways & Limitations
For non-unitary processes, fidelity measures do not have a simple relationship.
Abstract
from arXiv · showhide
Quantum error correction (QEC) is an essential step towards realising scalable quantum computers. Theoretically, it is possible to achieve arbitrarily long protection of quantum information from corruption due to decoherence or imperfect controls, so long as the error rate is below a threshold value. The two-dimensional surface code (SC) is a fault-tolerant error correction protocol} that has garnered considerable attention for actual physical implementations, due to relatively high error thresholds ~1%, and restriction to planar lattices with nearest-neighbour interactions. Here we show a necessary element for SC error correction: high-fidelity parity detection of two code qubits via measurement of a third syndrome qubit. The experiment is performed on a sub-section of the SC lattice with three superconducting transmon qubits, in which two independent outer code qubits are joined to a central syndrome qubit via two linking bus resonators. With all-microwave high-fidelity single- and two-qubit nearest-neighbour entangling gates, we demonstrate entanglement distributed across the entire sub-section by generating a three-qubit Greenberger-Horne-Zeilinger (GHZ) state with fidelity ~94%. Then, via high-fidelity measurement of the syndrome qubit, we deterministically entangle the otherwise un-coupled outer code qubits, in either an even or odd parity Bell state, conditioned on the syndrome state. Finally, to fully characterize this parity readout, we develop a new measurement tomography protocol to obtain a fidelity metric (90% and 91%). Our results reveal a straightforward path for expanding superconducting circuits towards larger networks for the SC and eventually a primitive logical qubit implementation.
Device fabrication
The device uses niobium CPW resonators and aluminum single-junction transmons fabricated on silicon. Figure 4 characterizes parity-check readout and conditioned non-nearest-neighbour entanglement.
- Device fabrication: The device is fabricated on a 720 µm silicon substrate with 200 nm sputtered niobium CPW resonators.Resonators are patterned by optical lithography and subtractive reactive-ion etching.
- Device fabrication: Three single-junction transmon qubits use aluminum layers of 35 nm and 85 nm formed by double-angle deposition and liftoff.
- Parity-check characterization: Figure 4 uses M2 histograms and conditioned Pauli state vectors to characterize parity readout and Bell-state generation between Q1 and Q3.The figure includes computational-basis inputs, equal-superposition inputs, and syndrome-conditioned tomography.
- Parity-check characterization: 0.90 and 0.91 are the reported measurement fidelities for the odd and even parity projections, respectively.The unconditional map has fidelity 0.968.
Device parameters
The three transmons operate near 5 GHz with dedicated readout resonators near 6.6–6.7 GHz and coherence times in the tens of microseconds. Two unmeasured bus resonators link adjacent qubits.
- Device parameters: The transmon transition frequencies are {5.0388, 5.0080, 5.2286} GHz, while readout resonators are at {6.698, 6.585, 6.695} GHz.
- Device parameters: The bus resonators are unmeasured and designed to resonate at 8 and 8.5 GHz.
- Device parameters: Readout-resonator coupling strengths are {70, 67, 67} MHz, with measured anharmonicities of −340 MHz for all qubits.
Experimental setup
The half-plaquette is operated in a dilution refrigerator with dedicated readout lines for each qubit. Q2 additionally uses a JPA, and all single-shot traces undergo optimal-quadrature filtering.
- Experimental setup: The device is cooled to 15 mK in an Oxford Triton dilution refrigerator.
- Experimental setup: Each qubit has a dedicated readout line with isolators and Caltech HEMT amplifiers, while Q2 uses an additional UC Berkeley JPA.
- Experimental setup: Microwave control signals use vector modulation, and readout pulses are generated by Arbitrary Pulse Sequencers.
- Experimental setup: All single-shot readout traces are processed with an optimal quadrature rotation filter.
Calibration sequences
Automated repeated-pulse sequences calibrate single-qubit and cross-resonance gates. ZX90 amplitude and phase errors are amplified with repetition, enabling precise and regularly refreshed calibration.
- Calibration sequences: Single-qubit gates are calibrated using automated repeated sequences described in prior work.
- Calibration sequences: For ZX90 amplitude calibration, an odd number 2N − 1 of pulses is applied and the amplitude is tuned toward a signal halfway between 0 and 1.Increasing N amplifies departures caused by amplitude miscalibration.
Randomized benchmarking
The experiment characterizes single- and two-qubit control using randomized benchmarking, including simultaneous operation of the two ZX90 links. These measurements establish gate fidelities and error estimates for the parity-check primitives.
- Single-qubit benchmarking: Single-qubit gates use calibrated Gaussian microwave pulses and are independently characterized with Clifford randomized benchmarking.The pulses include derivative-of-Gaussian quadrature shaping to reduce leakage.
- Single-qubit benchmarking: Table I summarizes the single-qubit randomized-benchmarking results.
- Two-qubit benchmarking: 350 ns ZX90 gates are shaped with Gaussian turn-on and turn-off segments surrounding a flat section.The two gates can be applied simultaneously because they commute.
- Two-qubit benchmarking: 3.8% and 4.3% estimated ZX90 gate errors are obtained for the Q1–Q2 and Q3–Q2 pairs, respectively.These estimates derive from two-qubit Clifford errors of 0.058 ± 0.003 and 0.065 ± 0.002, respectively, with each Clifford comprising 1.5 ZX90 generators.
Readout characterization
Readout characterization combines independently coupled measurement resonators with distinct readout methods for the code and syndrome qubits. The measured assignment fidelities and error ratios quantify channel performance and its fluctuations.
- Readout methods: Each qubit has its own measurement resonator, using high-power readout for Q1 and Q3 and JPA-assisted dispersive readout for Q2.Integration times are 4 µs for high-power readout and 2 µs for dispersive readout.
- Assignment performance: Assignment fidelities for the three readout channels are 0.84, 0.91 and 0.89, respectively.These typical values fluctuate by about 2–3% during a typical experiment.
- Experimental setup: Extended Data Figure 1 depicts the room-temperature wiring and internal dilution-refrigerator configuration.
- Assignment performance: The undesired-to-desired-state ratios are 9.9%(22%) for Q1, 5.7%(8.3%) for Q2, and 13.7%(6.0%) for Q3 under ground(excited) preparation.The ratios are obtained by fitting double-Gaussian models to the readout data.
State tomography
State tomography reconstructs the three-qubit state from correlated single-shot measurements after complete sets of post-rotations. Linear inversion and a constrained semidefinite program provide state estimates and fidelity uncertainty assessments.
- Reconstruction method: Correlated single shots from all measurement resonators are combined with complete post-rotations to reconstruct the state.Reconstructions use either linear inversion or a semidefinite program enforcing ρ ⪰ 0 and tr(ρ) = 1.
- Measurement protocol: Typically 20,000 shots are collected for each post-rotation.
- Fidelity estimation: State fidelity and statistical error are estimated from the reconstructed measurements using bootstrapping.The fidelity compares the ideal state with the reconstructed noisy state.
- Fidelity estimation: Statistical fluctuations are smaller than the difference between linear and semidefinite-program reconstructions, while the sum of negative eigenvalues is typically below 0.03.
Measurement tomography
The paper formulates syndrome-conditioned parity readout as two non-trace-preserving quantum operations and characterizes them through measurement tomography. Fidelity is evaluated using normalized outputs and averaging over random input states.
- Measurement tomography: Measurement tomography determines the conditional even- and odd-parity maps, which are completely positive but not trace-preserving.The maps are reconstructed separately after binning syndrome-qubit measurement results.
- Measurement tomography: The tomography protocol prepares diverse two-qubit input states and measurement bases, then reconstructs physical operations using linear reconstruction and minimization.The same pre- and post-rotation set used for state tomography supplies the required bases.
- Measurement tomography: The measurement fidelity normalizes each conditional output by its trace norm before comparing it with the ideal operation.For x = {even, odd}, the normalized output is x(ψ) = Ax(|ψ⟩⟨ψ|)/∥Ax(|ψ⟩⟨ψ|)∥tr.
- Measurement tomography: The reported fidelity is obtained by averaging over 150,000 random states drawn from the Fubini-Study measure.This averaging defines the operational fidelity metric for the noisy and ideal projections.
- Measurement tomography: A process fidelity based on normalized Choi matrices is also possible, but it has no simple relationship to standard fidelity for non-unitary processes.The unconditional map instead arises by tracing over the syndrome qubit, where standard quantum-operation fidelity applies.
NOTE
The authors report becoming aware of similar work by O. P. Saira et al. during manuscript completion. This places the study alongside contemporaneous related research.
- NOTE: During manuscript completion, the authors became aware of similar work by O. P. Saira et al.The passage identifies this study as contemporaneous related work.
- NOTE: O. P. Saira et al. are identified as the authors of the similar work.The passage gives the researchers’ names but no further comparison.
- NOTE: The related-work disclosure occurred during completion of the manuscript.No additional details about the other study are provided.