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High-Fidelity Single-Shot Toffoli Gate via Quantum Control
Ehsan Zahedinejad, Joydip Ghosh, Barry C. Sanders
TL;DR
Decomposed Toffoli gates are slower and have lower overall fidelity than a single-shot implementation, despite the gate's relevance to reversible computing and quantum error correction. This paper uses enhanced differential evolution for direct quantum control of a three-transmon Toffoli gate. The approach reports fidelity around 99.9% under realistic conditions and a gate time as fast as an entangling two-qubit gate.
Problem
Toffoli gates have been implemented through sequential single- and two-qubit gates, while non-topological quantum error correction requires an operation on at least three qubits.
Source: p. 1Method
The authors use a non-greedy quantum-control procedure based on an enhanced Differential Evolution algorithm to construct a single-shot Toffoli gate.
Source: pp. 1, 3Results
99.9% fidelity is reported for a three-qubit CCZ gate subject to realistic conditions, at a speed comparable to a two-qubit gate.
Source: pp. 2–3Takeaways & Limitations
A directly realized CCNOT can support high-fidelity error correction, and CCNOT together with Hadamard forms a universal gate set.
Source: p. 1The decoherence model assumes T1 = T2 and truncates phase-damping terms beyond Kraus index 3, with the truncation justified when gate time is much shorter than T2.
Source: p. 2, Decoherence · p. 8
Abstract
from arXiv · showhide
A single-shot Toffoli, or controlled-controlled-NOT, gate is desirable for classical and quantum information processing. The Toffoli gate alone is universal for reversible computing and, accompanied by the Hadamard gate, forms a universal gate set for quantum computing. The Toffoli gate is also a key ingredient for (non-topological) quantum error correction. Currently Toffoli gates are achieved by decomposing into sequentially implemented single- and two-qubit gates, which requires much longer times and yields lower overall fidelities compared to a single-shot implementation. We develop a quantum-control procedure to construct a single-shot Toffoli gate for three nearest-neighbor-coupled superconducting transmon systems such that the fidelity is 99.9% and is as fast as an entangling two-qubit gate under the same realistic conditions. The gate is achieved by a non-greedy quantum control procedure using our enhanced version of the Differential Evolution algorithm.
Evolution (SuSSADE)
SuSSADE adapts differential evolution by switching between subspace and whole-space breeding, with a one-dimensional subspace sufficient for this application. The section explains why standard and self-adaptive variants fall short and how candidate solutions are evolved.
Evolution (SuSSADE)
- Candidate elements encode piecewise-constant control functions, each held fixed for an interval Δτ = Θ/N.Θ is the total duration of the unitary operation, and N is the number of optimization parameters.Source: Evolution (SuSSADE) · p. 6
- DE evolves candidate solutions through mutation, crossover, and selection, with the objective function in this application being intrinsic fidelity.Mutation forms a donor vector; crossover exchanges its elements with a candidate, and selection chooses the next-generation solution by objective value.Source: Evolution (SuSSADE) · p. 6
- Standard DE did not exceed 95% Toffoli fidelity, while trial-and-error tuning of its parameters was computationally intractable for this problem.The objective-function evaluation cost and resources required to run DE for a long time made that tuning approach impractical.Source: Evolution (SuSSADE) · p. 6
- Self-adaptive DE remained below the approximately 99.9% threshold, and tested DECC-II combinations also failed to meet the required Toffoli fidelity.DECC-II divides candidate solutions into subspaces, but fidelity depended strongly on its user-chosen cycle count and subspace dimension.Source: Evolution (SuSSADE) · p. 7
- SuSSADE switches breeding between a selected subspace and the whole search space; choosing m = 1 suffices here and defines the 1DSUSSADE variant.It sets s = 1 and selects subspace dimension m from {1, 2, 3, 4, 5}; switching is intended to speed convergence over subspace-only optimization.Source: Evolution (SuSSADE) · p. 7
Phase compensation
The CCZ target is designed using phase compensation: phases that can be corrected with post-rotations about the z-axis need not be part of the target pulse. This lets the target accommodate those removable phases.
Phase compensation
- Phase compensation is used to design the optimal pulse for the CCZ gate.The approach is described as frequently used in designing quantum gates with superconducting atoms.Source: Phase compensation · p. 7
- Phases acquired by the |100⟩, |010⟩, and |001⟩ states are assumed nullifiable by post-rotations about the z-axis.The text notes that arbitrary z-axis rotations are trivial for a superconducting qubit and are usually performed via qubit-frequency excursions.Source: Phase compensation · p. 7
- The target gate may include additional phases that can be compensated by post-rotations about the z-axis.The ideal CCZ is defined as diag(1, 1, 1, 1, 1, 1, 1, −1).Source: Phase compensation · p. 7
Decoherence
Decoherence is modeled by evolving the system's density matrix under amplitude and phase damping. The calculation constructs the full system's Kraus operators from single-transmon operators and truncates the phase-damping series for short gate times.
Decoherence
- Each transmon is modeled as a damped harmonic oscillator, and amplitude- and phase-damping maps are combined as tensor products across transmons under an independence assumption.The density matrix evolves under Kraus operator-sum representations of the dissipative processes.Source: Decoherence · p. 7
- Amplitude-damping Kraus matrices are specified for a single transmon assuming four energy levels, with relaxation characterized by T1.The amplitude-damping factor is exp(−t/T1).Source: Decoherence · pp. 7–8
- Phase damping is truncated to Kraus indices 0 through 3, which the authors consider valid when gate time is much shorter than T2.This condition implies t/T2 ≪ 1; higher-order terms are neglected.Source: Decoherence · p. 8
Proposed experimental realization
The proposed experimental procedure accounts for pulse distortion and calibrates residual noise through closed-loop learning control. It tests optimized pulses on the noisy system and repeats optimization when measured fidelity misses the target.
Proposed experimental realization
- Square control pulses are smoothed by the superconducting control electronics, so the calibration procedure accounts for distortion of the designed pulses.The ideal no-noise pulse assumption does not hold in an experiment with pulse-distorting noise.Source: Proposed experimental realization · p. 8
- First-order distortion is modeled by convolving a square pulse with a Gaussian, yielding an error-function pulse already included in the optimal pulse design.Higher-order distortion requires calibration using closed-loop learning control.Source: Proposed experimental realization · p. 8
- Closed-loop learning control tests SuSSADE-generated pulses experimentally and reruns the optimization if measured fidelity misses the target.The process continues until the target is met; the schematic shows the pulses entering a noisy quantum system.Source: Proposed experimental realization · p. 8, Figure 4 · p. 8