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Quantum walks on a programmable two-dimensional 62-qubit superconducting processor
Ming Gong, Shiyu Wang, Chen Zha, Ming-Cheng Chen, He-Liang Huang, Yulin Wu, Qingling Zhu, Youwei Zhao, Shaowei Li, Shaojun Guo, Haoran Qian, Yangsen Ye, Fusheng Chen, Chong Ying, Jiale Yu, Daojin Fan, Dachao Wu, Hong Su, Hui Deng, Hao Rong, Kaili Zhang, Sirui Cao, Jin Lin, Yu Xu, Lihua Sun, Cheng Guo, Na Li, Futian Liang, V. M. Bastidas, Kae Nemoto, W. J. Munro, Yong-Heng Huo, Chao-Yang Lu, Cheng-Zhi Peng, Xiaobo Zhu, Jian-Wei Pan
TL;DR
Programmable two-dimensional quantum walks require controllable superconducting-qubit networks, but frequency-alignment optimization becomes impractical beyond 60 qubits. The paper fabricates and calibrates a 62-functional-qubit array to implement quantum walks and Mach-Zehnder interferometers. It demonstrates high-fidelity single- and two-walker evolution with disorder-controlled interference fringes and interacting two-walker behavior.
Problem
Programmable two-dimensional quantum walks require controllable superconducting-qubit networks, while frequency-alignment optimization becomes impractical beyond 60 qubits.
Method
The authors fabricate a 62-functional-qubit array, calibrate frequency disorder, and tune qubit frequencies to implement quantum walks and Mach-Zehnder interferometers.
Results
The processor demonstrates high-fidelity single- and two-walker quantum walks, coherent two-path propagation, and disorder-controlled interference fringes for single and double walkers.
Takeaways & Limitations
The work provides a basis for programmable two-dimensional quantum walks, Mach-Zehnder interferometers, and multi-walker hard-core-boson interference.
Takeaways & Limitations
Thermal noise during detuning operations affects the quality of the quantum walks.
Abstract
from arXiv · showhide
Quantum walks are the quantum mechanical analogue of classical random walks and an extremely powerful tool in quantum simulations, quantum search algorithms, and even for universal quantum computing. In our work, we have designed and fabricated an 8x8 two-dimensional square superconducting qubit array composed of 62 functional qubits. We used this device to demonstrate high fidelity single and two particle quantum walks. Furthermore, with the high programmability of the quantum processor, we implemented a Mach-Zehnder interferometer where the quantum walker coherently traverses in two paths before interfering and exiting. By tuning the disorders on the evolution paths, we observed interference fringes with single and double walkers. Our work is an essential milestone in the field, brings future larger scale quantum applications closer to realization on these noisy intermediate-scale quantum processors.
3 Shanghai Research Center for Quantum Sciences, Shanghai 201315, China
This passage gives an affiliation address in Atsugi, Kanagawa, Japan.
- NTT Basic Research Laboratories and the Research Center for Theoretical Quantum Physics are located in Atsugi, Kanagawa, Japan.
5 National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
The programmable 62-qubit processor realizes two-dimensional quantum walks and Mach-Zehnder interferometers with single and two walkers. Tunable qubit frequencies control propagation paths, disorder, and interference.
- Processor design: The 8×8 superconducting array contains 62 functional qubits and uses pass-through holes as an alternative wiring solution.Two qubits and one coupling resonator are non-functional.
- Continuous-time quantum walks: Continuous-time quantum walks were implemented by tuning qubits to a common interaction frequency and measuring all-qubit populations during evolution.The experiment used evolution times from 0 to 600 ns and 50,000 single-shot measurements per time point.
- Continuous-time quantum walks: 22.2 ± 2.0 site/µs is the measured propagation velocity, with the observed value limited by the Lieb-Robinson bound and attributed differences to short distance and disorder.The experiment determines velocity using a two-site correlation function.
- Mach-Zehnder interferometer: A Mach-Zehnder interferometer splits a walker into two spatially separated paths, reconnects them, and produces refocusing with population as high as 0.43 at t = 650 ns.The experimental evolution agrees well with numerical simulations.
- Mach-Zehnder interferometer: Changing path disorder produces interference fringes by modifying tunneling amplitudes and accumulated propagation phases, while blocking one path removes the fringes.The results require coherence across the spatially separated paths.
- Two-walker interferometry: Two-walker interference fringes arise from interactions between the walkers, consistent with hard-core-boson transmon physics in the |U/Jeff| ∼120 limit.Removing the alternate-path mechanism eliminates the fringes, and separate single-walker results do not reproduce their sum.
1 Experimental wiring setup
The 62-qubit processor uses dedicated control and readout wiring, with staged filtering and amplification to suppress noise and process signals.
- Each qubit has individual XY, fast Z, and DC control lines connected through bias tees.The lines support qubit driving, Z-pulse control, and DC biasing to idle points.
- Attenuators and low-pass filters are installed at multiple dilution-refrigerator stages to reduce thermal and control-line noise.The reported attenuations are 41 dB for XY, 34 dB for fast Z, and 61 dB for readout input lines.
- Room-temperature DACs and microwave sources generate qubit-control and readout pulses, while ADCs capture and analyze amplified readout signals.DACs also generate fast Z-control pulses, and DC sources bias qubits and JPAs.
- Readout signals are amplified first by JPAs, then by HEMT and room-temperature amplifiers before digitization and demodulation.Circulators and filters precede the amplification chain to block noise from higher-temperature stages.
2 Device design and fabrication
The device is an 8×8 superconducting transmon array built from modular four-qubit units, with nearest-neighbor resonator coupling and multilayer fabrication.
- Each modular unit contains four frequency-tunable transmons with individual readout resonators and a shared band-pass readout filter.Each qubit couples to four nearest neighbors through coplanar waveguide resonators, with approximately 4 mm qubit separation.
- The processor fabrication begins with a 100 nm aluminum film grown on a 2-inch sapphire wafer.Optical lithography and wet etching define resonators, filters, control lines, and transmon capacitors.
- Airbridge scaffolds and aluminum airbridges are fabricated before electron-beam lithography and double-angle evaporation form Al/AlOx/Al Josephson junctions.The sequence uses deposited SiO2 scaffolds followed by aluminum airbridges.
- The wafer is diced into a square chip, pass-through holes are laser-fabricated, and the chip is wire-bonded to a PCB.
3 Parameters of the superconducting quantum device
The processor’s qubit, coupling, and readout parameters are distributed across the array and support a hard-core-boson description with high state-readout fidelity.
- The 62 functional qubits occupy an 8×8 array of 16 units, with individually tunable transmon frequencies and DC-defined idle points.Maximum frequencies range from 5.087 to 5.711 GHz, while idle frequencies range from 4.912 to 5.620 GHz.
- The parameter maps report maximum and idle frequencies, anharmonicity, relaxation times, and dephasing times for each qubit.Each square represents one qubit, with its number and color encoding the corresponding parameter value.
- 2.01 MHz is the mean effective neighboring-qubit coupling at 5.02 GHz, giving |U/J| = 124 and a hard-core-boson description.
- 96.6% and 91.9% are the average readout fidelities for states |0⟩ and |1⟩, respectively.The same readout characterization includes drive frequency, qubit–resonator coupling, dispersive shift, and resonator linewidth distributions.
4 System calibration
The calibration procedure suppresses frequency disorders while avoiding defects, crosstalk, ZZ coupling, and readout-fidelity trade-offs. A separate two-step interferometer optimization balances the paths and increases population at site D.
- 4.1 Idle frequency setup: The idle-frequency setup avoids defect-affected frequencies, limits microwave crosstalk and ZZ coupling, and balances readout fidelity against frequency crowding.Nearest-neighbor crosstalk is around -25 dB, the minimum neighboring-qubit gap is 50 MHz, and the minimum idle frequency is 4.9 GHz.
- 4.1 Idle frequency setup: A three-step procedure measures frequency-dependent T1, selects available frequencies, and re-optimizes qubits whose performance changes at their idle points.Available-frequency tables use 1 MHz steps, and T1 weights the frequency choices toward better-performing points.
- 4.2 Frequency-alignment optimization: The alignment procedure uses multi-qubit swapping data and Nelder-Mead optimization to fit a disorder map and choose the lower-distance correction.The correction sign is selected by comparing two alignment setups through their overall distances.
- 4.2 Frequency-alignment optimization: The final maximal disorder is smaller than 0.8Jeff after repeated alignment updates.The overall distance saturates after several rounds of multi-qubit swapping and correction.
- 4.3 Optimization of the interferometer: Residual interferometer disorders initially limit site-D population to 0.12 at t = 650 ns, motivating a two-step optimization that avoids blocking one path.The first step balances the two paths by optimizing the product of the populations at sites L10 and R10.
- 4.3 Optimization of the interferometer: Above 0.43 population at site D at t = 650 ns is achieved after the two-step interferometer optimization.The procedure first balances the paths, then optimizes the remaining alignment frequencies.
5 Thermalization and Post-selection
Thermal noise during detuning affects quantum-walk quality, while post-selection suppresses thermalization by retaining measurements with conserved total excitation number. In the two-walker data, 11.3% of single-shot measurements were retained.
- Thermalization: Thermal noise is non-negligible for several qubits during detuning operations and affects quantum-walk quality.The authors suggest heating of bonding wires to control-line electrodes as a possible cause.
- Post-selection: Post-selection retains events whose total recorded excitation number equals the total initial excitation number.This technique is used to partially remove circuit-loss and thermalization effects.
- Post-selection: 11.3% of all single-shot measurements are retained after post-selection in the two-walker data.The post-selected behavior is reported as much closer to ideal because thermalization is suppressed.
6 Numerical simulation method
The simulations approximate the hard-core Bose-Hubbard model with a spin model and truncate the Hilbert space to states matching the initial excitation number. This makes single- and two-excitation evolution computationally tractable.
- Model and truncation: The 62-qubit hard-core Bose-Hubbard evolution is simulated under a spin model using a fixed-excitation subspace.This approximation is considered reasonable when |U| ≫ |J| and the filling factor is low.
- Model and truncation: Truncation reduces the dimensions to 62 for one excitation and 1891 for two excitations, making the matrices manageable on a laptop.The full Hilbert space is beyond the capability of a state-of-the-art classical supercomputer.
- Time evolution: The time-independent Hamiltonian evolves the truncated initial state, allowing calculation of populations and correlation functions over time.The evolution uses |Ψ(t)⟩ = e^-iHt/ħ|Ψ(0)⟩.
7 The velocity of the correlations
Correlation propagation along the array diagonal is measured using two-site correlations and Gaussian-fitted propagation fronts. The measured velocity is below the Lieb-Robinson bound, while simulations show velocity increasing toward that bound with distance.
- Measurement: The experiment measures two-site correlations between the initial excitation and diagonal qubits to determine propagation velocity.The diagonal distance is defined using the Euclidean separation between sites.
- Experimental velocity: 22.2 ± 2.0 site/µs is the measured propagation velocity, compared with the Lieb-Robinson bound vmax = 35.7 site/µs.Gaussian fits identify propagation fronts, which are linearly fitted against distance.
- Distance dependence: Under 1.6 MHz random disorder, instantaneous velocity increases with distance and approaches vmax.The simulations use a 15×15-qubit array and extract velocity from local linear fits through signal positions.
- Distance dependence: 24.9 ± 5.2 site/µs at short distance agrees with the experimental velocity, whereas 35.0 ± 1.8 site/µs is reached at longer distance.The authors attribute the reduced experimental velocity to short distance and disorder.
8 The effect of random disorders and decoherence
Numerical simulations attribute discrepancies between experiment and simulation to random disorders and decoherence. Increasing disorder lowers the population reaching site D, while dephasing is also significant on the device timescale.
- Random disorders: Increasing random disorder decreases the maximum population at site D by causing reflection during traversal.The simulation detunes each path site's initial frequency by a random value between -RD and RD.
- Decoherence: The decoherence simulation used an 8-qubit substitute because the complete multi-qubit system was too large to simulate.The supplied passage states that three decoherence conditions were considered, but does not provide their complete list.
- Dephasing: At approximately 500 ns, population at D decays from 0.9 to 0.6 when the dephasing time is 1.6 µs.The average dephasing time is one order of magnitude lower than the energy-relaxation time and comparable to the evolution time.
9 Extended data
Extended data document quantum-walk dynamics, fidelity analysis, blocked-path controls, and single- and double-walker interferometer behavior through experiments, simulations, figures, and online movies. These materials include evidence of interference fringes associated with two-walker interaction.
- Quantum-walk dynamics: Supplementary figures show single-particle walks initialized at U00Q0 and U33Q2, alongside corresponding experimental and simulated population dynamics.The populations are illustrated at t = 0 ns, 100 ns, 200 ns, and 300 ns.
- Online extended data: Online movies provide time-resolved population dynamics for two-dimensional walks and single- and two-walker Mach–Zehnder interferometers.Movies S1 and S2 cover two- and single-walker quantum walks, while Movies S3 and S4 cover the interferometers.
- Fidelity: Fidelities of single- and double-particle quantum walks are plotted as functions of time using squared statistical overlap between measured and simulated distributions.The experimentally determined and numerically simulated population distributions are denoted p(i,j) and q(i,j), respectively.
- Control measurements: A blocked-path control shows the time evolution of populations when the {R} path is blocked in the single-particle interferometer.The qubit is excited at source S, and experimental and simulated populations are compared.
- Two-particle controls: Two-particle supplementary data compare interferometer configurations with and without sites BS1 and S, using experiments and numerical simulations.The sites are removed in one configuration to stop both walkers from back propagating along their alternate paths.
- Two-walker interference: Experimental and simulated difference plots show obvious interference fringes, indicating interaction between two walkers in the interferometer.The experimental difference compares the two-walker result with the sum of the corresponding single-walker results.