Protec'ng quantum gates from control noise Constantin Brif - - PowerPoint PPT Presentation
Protec'ng quantum gates from control noise Constantin Brif Sandia National Laboratories Collaborators: Matthew Grace and Kevin Young (Sandia) David Hocker, Katharine Moore,Tak-San Ho, and Herschel Rabitz (Princeton
Sandia National Laboratories is a multi-program laboratory managed and
Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.
A unitary quantum gate is the basic functioning element of a quantum circuit. Some basic notation: number of qubits in the quantum gate system dimension of the system’s Hilbert space the target unitary transformation the actual evolution operator at the final time T The same unitary transformation is applied to any input state:
An external classical control c(t) is necessary to operate the quantum gate. The Hamiltonian and evolution operator are functionals of the control: The gate fidelity is a measure of how well the target transformation is performed: It is convenient to use a normalized fidelity:
The gate fidelity is also a functional of the control:
The functional dependence F = F[c(t)] is called the control landscape. Critical points of the control landscape satisfy: A sufficient condition for
is negative semidefiniteness
For a recent review, see
New J. Phys. 12, 075008 (2010)
The analysis of regular critical points on the control landscape reveals that:
l There is one maximum manifold: F = 1 l There is one minimum manifold: F = 0 l All other critical manifolds are saddles
(can be avoided by a smart optimization algorithm) An optimal control solution c0(t) is perfect in ideal conditions: For quantum gate control with linear field coupling: The Hessian at the maximum: The Hessian at any optimal control solution has only non-positive eigenvalues. The “flatness” of the control landscape in the vicinity of an optimal solution depends on the number and magnitude of negative Hessian eigenvalues.
All actual controls are noisy! Control noise Gate errors F < 1 Consider a quantum gate that operates in the vicinity of an optimal control:
l additive noise: l multiplicative noise:
Expanding for small noise: We consider a random noise process, so the error z(t) is a stochastic variable with the autocorrelation function: Expected value of the quantum gate fidelity:
White noise has zero correlation time: Expected value of the quantum gate fidelity: The diagonal elements of the optimal Hessian are time independent: The general expression for the expected value of the quantum gate fidelity in the presence of small white noise:
Expected gate error is proportional to the total control time T:
For fidelities above 0.99, the discrepancy between the perturbative result and Monte Carlo averaging (over a sample of 20,000 noisy controls) is within the statistical error due to the finite sample size (~0.7%). The accuracy of the perturbative approximation is excellent for all fidelities above 0.9.
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Control time Expected gate error σa
2 = 10−1
σa
2 = 10−2
σa
2 = 10−3
σa
2 = 10−4
σa
2 = 10−5
σa
2 = 10−6
Expected gate error is proportional to the control fluence (energy):
The perturbative result is once again in excellent agreement with Monte Carlo averaging (over a sample of 20,000 noise realizations) for all gate fidelities above 0.9.
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10 Fluence of the optimal control field Expected gate error σm
2 = 10−1
σm
2 = 10−2
σm
2 = 10−3
σm
2 = 10−4
σm
2 = 10−5
σm
2 = 10−6
Optimizing robustness to additive white noise in controls is equivalent to minimizing the total control time. For a given system Hamiltonian H and target gate W, there exists a critical value T* of control time, below which the target is no longer reachable. We developed a numerical procedure to identify T* and explore the Pareto front for two competing control objectives: gate fidelity maximization and control time minimization.
Time-fidelity Pareto fronts for the CNOT gate in a two-qubit system, explored starting from different initial values of T. For control times above T*, it is possible to decrease T (and thus improve robustness) while keeping the nominal gate fidelity at 1. However, below T*, the nominal fidelity rapidly deteriorates as T decreases, i.e., one has to sacrifice fidelity to improve robustness.
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T
D (normalized)
T=6 T=4.2 T=4.4 T=4.3 T=4.6 T=4.5 T=5 T=4.4 T=4.8 T=4.2
T*
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W = SWAP, φ = 0 W = QFT, φ = 0 W = SWAP, φ = π/2 W = QFT, φ = π/2
−1 −0.5 0.5 1 1.5 2 2.5 −1 −0.5 0.5 1 1.5 CNOT QFT’ SWAP random identity QFT
log10(coupling strength J
(1,2))
log10(T*)
Time-fidelity Pareto fronts for different target gates (SWAP and QFT) and different values of the gate’s global phase (0 and π/2). The critical value T* of the control time depends both on the target gate and on its global phase. The critical value T* of the control time also depends on the Hamiltonian of the controlled system, in particular, on the strength of inter-qubit couplings. In the weak coupling regime, the dependence of the critical time on the coupling strength is
Optimizing robustness to multiplicative white noise in controls is equivalent to minimizing the fluence of the control field. Exploring the dependence of the optimal-field fluence on the initial-field fluence: Average fluence of 25 optimal control fields obtained by starting from random initial fields with the same fluence (the target gate is the Hadamard transform for a one-qubit system):
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Initial field fluence Average optimal field fluence T = 1 T = 3 T = 4 T = 6 T = 15 T = 20
To enhance the robustness to MWN, one should use initial fields with very small fluence.
The minimum value of the optimal-field fluence decreases with T, but not monotonically. Exploring the dependence of the minimum optimal-field fluence on control time T: Target: Hadamard transform Target: Pauli Y-gate
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Control time Fluence of the optimal control field
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Control time Fluence of the optimal control field
The envelope function decays as 1/T The amplitude of oscillations is constant The envelope function decays as 1/T The amplitude of oscillations decays as exp(-γT)
For small colored noise, the expected gate error (i.e., the expected fidelity decrease) is determined by an overlap integral involving the Hessian of the optimal control field and the noise autocorrelation function. We define this overlap as the robustness metric K. In particular, for additive colored noise, the robustness metric is For a wide-sense-stationary noise process: The Wiener-Khinchin theorem relates autocorrelation function and power spectral density: Thus, the robustness metric can be expressed as an overlap in the frequency domain: The goal is to minimize the expected gate error by searching for optimal controls with the Hessian “orthogonal” to the control noise (i.e., using the null space of the Hessian).
We search for optimal controls which minimize the robustness metric K, using a genetic algorithm (GA). For each control field (population member), we find its optimal counterpart using a gradient-based algorithm, compute its Hessian, and evaluate K. Control field parameterization used in GA optimization is an important constraint. More freedom in the control field yields better optimization results for the robustness. Robustness optimization results
(with ω = 20) and the noise autocorrelation function (with α = 20). The target gate is the Hadamard transform.
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Generation Robustness metric, K
field parameterization with 1 frequency field parameterization with 4 frequencies
expanded into the Taylor series around the optimal control solution.
function of the noise.
for all gate fidelities of interest for realistic quantum computing (above 0.9).
§ For additive white noise, the expected gate error is proportional to the
control time T.
u Control time can be decreased without sacrificing fidelity only above
the critical time T*.
§ For multiplicative white noise, the gate error is proportional to the control
fluence (energy).
u Optimal-field fluence decreases with control time as 1/T.
the overlap integral quantifying the expected gate error.