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arXiv:quant-ph/0502104 · Physical Review A

Optimal control-based efficient synthesis of building blocks of quantum algorithms: A perspective from network complexity towards time complexity

Thomas Schulte‐Herbrüggen, Andreas K. Spörl, Navin Khaneja, Steffen J. Glaser

Published Oct 27, 2005• 158 citations
arXivHTMLPDF

Abstract

In this paper, we demonstrate how optimal control methods can be used to speed up the implementation of modules of quantum algorithms or quantum simulations in networks of coupled qubits. The gain is most prominent in realistic cases, where the qubits are not all mutually coupled. Thus the shortest times obtained depend on the coupling topology as well as on the characteristic ratio of the time scales for local controls vs nonlocal (i.e., coupling) evolutions in the specific experimental setting. Relating these minimal times to the number of qubits gives the tightest known upper bounds to the actual time complexity of the quantum modules. As will be shown, time complexity is a more realistic measure of the experimental cost than the usual gate complexity. In the limit of fast local controls (as, e.g., in NMR), time-optimized realizations are shown for the quantum Fourier transform (QFT) and the multiply controlled NOT gate $({\mathrm{C}}^{n\ensuremath{-}1}\mathrm{NOT})$ in various coupling topologies of $n$ qubits. The speed-ups are substantial: in a chain of six qubits the quantum Fourier transform so far obtained by optimal control is more than eight times faster than the standard decomposition into controlled phase, Hadamard and SWAP gates, while the ${\mathrm{C}}^{n\ensuremath{-}1}\mathrm{NOT}$ gate for a completely coupled network of six qubits is nearly seven times faster.

DOI
10.1103/physreva.72.042331

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In this paper, we demonstrate how optimal control methods can be used to speed up the implementation of modules of quantum algorithms or quantum simulations in networks of coupled qubits. The gain is most prominent in realistic cases, where the qubits are not all mutually coupled. Thus the shortest times obtained depend on the coupling topology as well as on the characteristic ratio of the time scales for local controls vs nonlocal (i.e., coupling) evolutions in the specific experimental setting. Relating these minimal times to the number of qubits gives the tightest known upper bounds to the actual time complexity of the quantum modules. As will be shown, time complexity is a more realistic measure of the experimental cost than the usual gate complexity. In the limit of fast local controls (as, e.g., in NMR), time-optimized realizations are shown for the quantum Fourier transform (QFT) and the multiply controlled NOT gate (Cn\ensuremath−1NOT)({\mathrm{C}}^{n\ensuremath{-}1}\mathrm{NOT})(Cn\ensuremath−1NOT) in various coupling topologies of nnn qubits. The speed-ups are substantial: in a chain of six qubits the quantum Fourier transform so far obtained by optimal control is more than eight times faster than the standard decomposition into controlled phase, Hadamard and SWAP gates, while the Cn\ensuremath−1NOT{\mathrm{C}}^{n\ensuremath{-}1}\mathrm{NOT}Cn\ensuremath−1NOT gate for a completely coupled network of six qubits is nearly seven times faster.
Timeoptimal quantum computing provides a dramatic speed‐up in realistic coupling networks of qubits, wherever the time‐limiting steps are coupling evolutions, while local controls are fast. Already in a chain of six spin qubits the timeoptimal quantum Fourier transform obtained by optimal control is some six times faster than the standard decomposition into controlled phase and Hadamard gates. Here, new results of quantum control are presented within their Lie‐theoretical framework sketched from the preconditions of controllability and reachability up to recent methods of gradient flows.
Quantum Fourier transform (QFT) is a quantum analogue of the classical discrete Fourier transform. It is a fundamental quantum gate in quantum algorithms which has an exponential advantage over the classical computation and has been excessively studied. Normally, an n-qubit quantum Fourier transform could be resolved into the tensor product of n single-qubit operations, and each operation could be implemented by a Hadamard gate and a controlled phase gate. Then the complexity of an n-qubit QFT is of order O(n2). To reduce the complexity of quantum operations, optimal control (OC) method has recently been used successfully to find the minimum time for implementing a quantum operation. Up to now, two types of quantum optimal control methods have been presented, i.e. analytical and numerical methods. The analytical approach is to change the problem of efficient synthesis of unitary transformations into the geometrical one of finding the shortest paths. Numerical optimal control procedures are based on the gradient methods (GRAPE, Gradient Ascent Pulse Engineering) and Krotov methods. Notable application mainly focus on nuclear magnetic resonance fields, including imaging, liquid-state NMR, solid-state NMR, and NMR quantum computation. One obvious advantage of optimal control NMR quantum computation is that the OC unitary evolution transformation pulse sequences are normally shorter than the conventional corresponding ones. Here we use the optimal control method to find the minimum duration for implementing QFT quantum gate. A linear spin chain with nearest-neighbor Ising interaction is used to find the optimization. And the optimized pulse sequence is experimentally demonstrated on an NMR quantum information processor. By using optimal control method with numerical calculation, a three-qubit QFT in an indirect-linear-coupling chain system is optimized. The duration of the OC QFT is obviously shorter than that of conventional approaches. The OC pulse sequence has been experimentally implemented on a liquid-state NMR spectrometer. To verify the optimally controlled pulse sequence for the three-qubit QFT, different initial states are assumed. The accuracy of the OC pulse sequence could be demonstrated by the consistency of theoretical simulation spectra with the experimental results. The good consistency between the simulation and the experimental spectra demonstrates that the OC QFT is of high fidelity.
The Quantum Fourier transform (QFT) is a key ingredient in most quantum algorithms. We have compared various spin-based quantum computing schemes to implement the QFT from the point of view of their actual time-costs and the accuracy of the implementation. We focus here on an interesting decomposition of the QFT as a product of the non-selective Hadamard transformation followed by multiqubit gates corresponding to square- and higher-roots of controlled-NOT gates. This decomposition requires only O(n) operations and is thus linear in the number of qubits n. The schemes were implemented on a two-qubit NMR quantum information processor and the resultant density matrices reconstructed using standard quantum state tomography techniques. Their experimental fidelities have been measured and compared.
The Quantum Fourier transform (QFT) is a key ingredient in most quantum algorithms. We have compared various spin-based quantum computing schemes to implement the QFT from the point of view of their actual time-costs and the accuracy of the implementation. We focus here on an interesting decomposition of the QFT as a product of the non-selective Hadamard transformation followed by multiqubit gates corresponding to square- and higher-roots of controlled-NOT gates. This decomposition requires only O(n) operations and is thus linear in the number of qubits n. The schemes were implemented on a two-qubit NMR quantum information processor and the resultant density matrices reconstructed using standard quantum state tomography techniques. Their experimental fidelities have been measured and compared.
In this paper, we study time-optimal control problems related to a system of two coupled qubits where the time scales involved in performing unitary transformations on each qubit are significantly different. In particular, we address the case where unitary transformations produced by evolutions of the coupling take a much longer time as compared to the time required to produce unitary transformations on the first qubit, but a much shorter time as compared to the time to produce unitary transformations on the second qubit. We present a canonical decomposition of SU(4) in terms of the subgroup $\text{SU}(2)\ifmmode\times\else\texttimes\fi{}\text{SU}(2)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$, which is natural in understanding the time-optimal control problem of such a coupled qubit system with significantly different time scales. A typical setting involves dynamics of a coupled electron-nuclear spin system in pulsed electron paramagnetic resonance experiments at high fields. Using the proposed canonical decomposition, we give time-optimal control algorithms to synthesize various unitary transformations of interest in coherent spectroscopy and quantum-information processing.
In two and three coupled Josephson charge qubits, we exemplify how to take pulses for realizing quantum gates from fidelity-limited pioneering stages to the decoherence limit of near time optimal high-fidelity controls. Thus, a CNOT gate can be obtained with a fidelity >1>1>1--${10}^{\ensuremath{-}9}$ for the two qubits. Even when including higher charge states, the leakage is below 11%1, although the pulses are nonadiabatic. The controls are five times faster than the pioneering experiment [Nature (London) 425, 941 (2003)] for otherwise identical parameters---i.e., a progress towards the error-correction threshold by a factor of 100. We outline schemes to generate these shaped pulses by Cauer synthesis, or more generally by few LCR circuits. The approach generalizes to larger systems: e.g., directly realizing a TOFFOLI gate in three linearly coupled charge qubits is shown to be 13 times faster than decomposing it into a circuit of nine CNOT gates of the above experimental work. In view of the next generation of fast pulse shapers, the combination of methods is designed to find wide application in quantum control of pseudospin and macroscopic quantum systems.