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Demonstration of Two-Qubit Algorithms with a Superconducting Quantum Processor
L. DiCarlo, J. M. Chow, J. M. Gambetta, Lev S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, R. J. Schoelkopf
TL;DR
The paper demonstrates Grover and Deutsch–Jozsa algorithms on a two-qubit processor and evaluates their reconstructed output states. Grover’s algorithm reaches 85% fidelity to the correct answer, while the results suggest both algorithms could exceed the 50% classical success probability with single-shot readout.
Problem
The work addresses whether a two-qubit processor can execute Grover and Deutsch–Jozsa algorithms with outputs sufficiently faithful to their ideal answers.
Method
The processor executes gate sequences, uses oracle phase inversion and entangling operations, and reconstructs output states through quantum state tomography.
Results
85% fidelity was obtained for Grover’s reconstructed output state, and the reported fidelities suggest both algorithms could exceed 50% classical success with single-shot readout.
Takeaways & Limitations
The reported processor results support using these two algorithms as demonstrations of quantum computation beyond the best classical single-oracle success probability.
Abstract
from arXiv · showhide
By harnessing the superposition and entanglement of physical states, quantum computers could outperform their classical counterparts in solving problems of technological impact, such as factoring large numbers and searching databases. A quantum processor executes algorithms by applying a programmable sequence of gates to an initialized register of qubits, which coherently evolves into a final state containing the result of the computation. Simultaneously meeting the conflicting requirements of long coherence, state preparation, universal gate operations, and qubit readout makes building quantum processors challenging. Few-qubit processors have already been shown in nuclear magnetic resonance, cold ion trap and optical systems, but a solid-state realization has remained an outstanding challenge. Here we demonstrate a two-qubit superconducting processor and the implementation of the Grover search and Deutsch-Jozsa quantum algorithms. We employ a novel two-qubit interaction, tunable in strength by two orders of magnitude on nanosecond time scales, which is mediated by a cavity bus in a circuit quantum electrodynamics (cQED) architecture. This interaction allows generation of highly-entangled states with concurrence up to 94%. Although this processor constitutes an important step in quantum computing with integrated circuits, continuing efforts to increase qubit coherence times, gate performance and register size will be required to fulfill the promise of a scalable technology.
A. Device fabrication
The processor combines superconducting transmons, cavity control, flux pulses, and state tomography to execute Grover’s search. The implementation reaches the correct output with 85% fidelity while demonstrating intermediate superposition and entanglement.
- Device fabrication: A 180 nm Nb film and patterned coplanar-waveguide structures form the cavity and flux-bias hardware, while Al transmon features are fabricated on separate chips.The transmons use interdigitated capacitors and split junctions fabricated by electron-beam lithography, double-angle evaporation, oxidation, and lift-off.
- Device operation: The device is cooled to 13 mK in a 3He-4He dilution refrigerator for processor operation.
- Grover implementation: State tomography shows simultaneous rotation into a maximal superposition, phase marking of |1, 0⟩, Bell-state entanglement, and subsequent disentangling.The final rotations produce the computational-basis output.
- Grover performance: 85% fidelity is obtained for the final Grover output state |1, 0⟩.
B. cQED Theory
The cQED theory models the tunable cavity-mediated interaction and its experimentally constrained parameters. The resulting processor supports algorithmic performance measurements summarized for Grover and Deutsch–Jozsa execution.
- Hamiltonian model: The generalized Tavis–Cummings Hamiltonian describes multi-level transmon qubits coupled through a microwave cavity.
- Parameterization: The model makes qubit transition frequencies and level-dependent couplings functions of charging energy, Josephson energy, and flux control.Flux control enters through the qubit Josephson energy and a linear flux-voltage relation including crosstalk and offsets.
- Parameter fitting: Spectroscopy and transmission data constrain simultaneous numerical fits using five transmon levels and five cavity-photon levels.
- Device parameters: The cavity linewidth is κ/2π = 1 MHz, with fitted couplings gL(R)/2π = 199 (183) MHz.
- Algorithmic performance: Table I summarizes the reconstructed output-state fidelities for the Grover and Deutsch–Jozsa algorithms.The reported uncertainties are based on 10 or 8 repetitions.
A. Perturbation Theory
Perturbative analysis explains the cavity-mediated frequency shift and its large on-off ratio near higher-level transmon resonances. Comparison with numerical diagonalization identifies where the three-level approximation is reliable.
- Perturbative analysis: Perturbation theory analyzes the large on-off ratio of the frequency shift ζ using a rotating-wave approximation and a truncation at three transmon excitations.
- Resonance condition: The perturbative expression diverges when one transmon’s 0↔1 transition aligns with the other transmon’s 1↔2 transition.
- Model comparison: The three-level expression agrees reasonably with numerical diagonalization away from the divergence, whereas the two-level expression has incorrect magnitude and sign.
B. State Tomography
Two-qubit state tomography reconstructs the density matrix from expectation values of linearly independent measurement operators, using joint cavity readout and constrained maximum-likelihood estimation.
- Tomographic reconstruction: A two-qubit density matrix ρ is decomposed using 16 linearly independent operators, whose coefficients are estimated from measured expectation values.Trace normalization reduces the required measurements to 15 independent operators.
- Measurement: Joint dispersive cavity readout measures one- and two-qubit Pauli operators, enabling access to correlations through calibrated quadrature coefficients.For this experiment, τ = 450 ns and (β1, β2, β12) ≈ (60, 50, 40) µV.
- Measurement: Fifteen prerotations formed from single-qubit rotations generate a complete set of linearly independent measurement operators before readout.The measurement set exploits the presence of both one- and two-qubit operators in the cavity signal.
- Estimation: 450,000 repetitions per operator provide experimental averages, which are fit with maximum likelihood while enforcing hermiticity and positive semidefiniteness of ρ.The estimator incorporates physical-state constraints that direct inversion would not automatically respect.