We present the construction and detailed analysis of highly optimized self-refocusing pulse shapes for several rotation angles. We characterize the constructed pulses by the coefficients appearing in the Magnus expansion up to second order. This allows a semianalytical analysis of the performance of the constructed shapes in sequences and composite pulses by computing the corresponding leading-order error operators. Higher orders can be analyzed with the numerical technique suggested by us previously. We illustrate the technique by analyzing several composite pulses designed to protect against pulse amplitude errors, and on decoupling sequences for potentially long chains of qubits with on-site and nearest-neighbor couplings.We present the construction and detailed analysis of highly optimized self-refocusing pulse shapes for several rotation angles. We characterize the constructed pulses by the coefficients appearing in the Magnus expansion up to second order. This allows a semianalytical analysis of the performance of the constructed shapes in sequences and composite pulses by computing the corresponding leading-order error operators. Higher orders can be analyzed with the numerical technique suggested by us previously. We illustrate the technique by analyzing several composite pulses designed to protect against pulse amplitude errors, and on decoupling sequences for potentially long chains of qubits with on-site and nearest-neighbor couplings.

Feb 1, 20240 citations

Scalable Design of Tailored Soft Pulses for Coherent Control

Pinaki Sengupta, Leonid P. Pryadko

We present a scalable scheme to design optimized soft pulses and pulse sequences for coherent control of interacting quantum many-body systems. The scheme is based on the cluster expansion and the time-dependent perturbation theory implemented numerically. This approach offers a dramatic advantage in numerical efficiency, and it is also more convenient than the commonly used Magnus expansion, especially when dealing with higher-order terms. We illustrate the scheme by designing 2nd-order self-refocusing pi pulses and a 6th-order 8-pulse refocusing sequence for a chain of qubits with nearest-neighbor couplings. We also discuss the performance of soft-pulse refocusing sequences in suppressing decoherence due to low-frequency environment.We present a scalable scheme to design optimized soft pulses and pulse sequences for coherent control of interacting quantum many-body systems. The scheme is based on the cluster expansion and the time-dependent perturbation theory implemented numerically. This approach offers a dramatic advantage in numerical efficiency, and it is also more convenient than the commonly used Magnus expansion, especially when dealing with higher-order terms. We illustrate the scheme by designing 2nd-order self-refocusing pi pulses and a 6th-order 8-pulse refocusing sequence for a chain of qubits with nearest-neighbor couplings. We also discuss the performance of soft-pulse refocusing sequences in suppressing decoherence due to low-frequency environment.

arXiv:quant-ph/0502029Physical Review LettersJul 15, 200535 citations

Soft-pulse dynamical decoupling with Markovian decoherence

Leonid P. Pryadko, Gregory Quiroz

We consider the effect of broadband decoherence on the performance of refocusing sequences, having in mind applications of dynamical decoupling in concatenation with quantum error correcting codes as the first stage of coherence protection. Specifically, we construct cumulant expansions of effective decoherence operators for a qubit driven by a pulse of a generic symmetric shape and for several sequences of $\ensuremath{\pi}$ and $\ensuremath{\pi}/2$ pulses. While, in general, the performance of soft pulses in decoupling sequences in the presence of Markovian decoherence is worse than that of the ideal $\ensuremath{\delta}$ pulses, it can be substantially improved by shaping.We consider the effect of broadband decoherence on the performance of refocusing sequences, having in mind applications of dynamical decoupling in concatenation with quantum error correcting codes as the first stage of coherence protection. Specifically, we construct cumulant expansions of effective decoherence operators for a qubit driven by a pulse of a generic symmetric shape and for several sequences of $\ensuremath{\pi}$ and $\ensuremath{\pi}/2$ pulses. While, in general, the performance of soft pulses in decoupling sequences in the presence of Markovian decoherence is worse than that of the ideal $\ensuremath{\delta}$ pulses, it can be substantially improved by shaping.

arXiv:0907.1315Physical Review AOct 20, 200923 citations

Ultrahigh-fidelity composite rotational quantum gates

Hayk L. Gevorgyan, Nikolay V. Vitanov

Composite pulse sequences, which produce arbitrary predefined rotations of a qubit on the Bloch sphere, are presented. The composite sequences contain up to 17 pulses and can compensate up to 8 orders of experimental errors in the pulse amplitude and the pulse duration. Composite sequences for three basic quantum gates, X (not), Hadamard and arbitrary rotation, are derived. Three classes of composite sequences are presented: one symmetric and two asymmetric. They contain as their lowest members two well-known composite sequences: the three-pulse symmetric SCROFULOUS pulse and the four-pulse asymmetric BB1 pulse, which compensate first- and second-order errors, respectively. The shorter sequences are derived analytically, and the longer ones numerically (instead by nesting and concatenation, as usually done hitherto). Consequently, the composite sequences derived here match or outperform the existing ones in terms of either speed or accuracy, or both. For example, we derive a second-order composite sequence, which is faster (by about 13%) than the famous BB1 sequence. For higher-order sequences, the speedup becomes much more pronounced. This is important for quantum information processing as the sequences derived here provide more options for finding the sweet spot between ultrahigh fidelity and high speed.Composite pulse sequences, which produce arbitrary predefined rotations of a qubit on the Bloch sphere, are presented. The composite sequences contain up to 17 pulses and can compensate up to 8 orders of experimental errors in the pulse amplitude and the pulse duration. Composite sequences for three basic quantum gates, X (not), Hadamard and arbitrary rotation, are derived. Three classes of composite sequences are presented: one symmetric and two asymmetric. They contain as their lowest members two well-known composite sequences: the three-pulse symmetric SCROFULOUS pulse and the four-pulse asymmetric BB1 pulse, which compensate first- and second-order errors, respectively. The shorter sequences are derived analytically, and the longer ones numerically (instead by nesting and concatenation, as usually done hitherto). Consequently, the composite sequences derived here match or outperform the existing ones in terms of either speed or accuracy, or both. For example, we derive a second-order composite sequence, which is faster (by about 13%) than the famous BB1 sequence. For higher-order sequences, the speedup becomes much more pronounced. This is important for quantum information processing as the sequences derived here provide more options for finding the sweet spot between ultrahigh fidelity and high speed.

arXiv:2012.14692Physical Review AJul 21, 202134 citations

Soft-pulse dynamical decoupling in a cavity

Leonid P. Pryadko, Gregory Quiroz

Dynamical decoupling is a coherent control technique where the intrinsic and extrinsic couplings of a quantum system are effectively averaged out by application of specially designed driving fields (refocusing pulse sequences). This entails pumping energy into the system, which can be especially dangerous when it has sharp spectral features like a cavity mode close to resonance. In this work we show that such an effect can be avoided with properly constructed refocusing sequences. To this end we construct the average Hamiltonian expansion for the system evolution operator associated with a single ``soft'' $\ensuremath{\pi}$ pulse. To second order in the pulse duration, we characterize a symmetric pulse shape by three parameters, two of which can be made zero by shaping. We express the effective Hamiltonians for several pulse sequences in terms of these parameters and use the results to analyze the structure of error operators for a controlled Jaynes-Cummings Hamiltonian. When errors are cancelled to second order, numerical simulations show excellent qubit fidelity with strongly suppressed oscillator heating.Dynamical decoupling is a coherent control technique where the intrinsic and extrinsic couplings of a quantum system are effectively averaged out by application of specially designed driving fields (refocusing pulse sequences). This entails pumping energy into the system, which can be especially dangerous when it has sharp spectral features like a cavity mode close to resonance. In this work we show that such an effect can be avoided with properly constructed refocusing sequences. To this end we construct the average Hamiltonian expansion for the system evolution operator associated with a single ``soft'' $\ensuremath{\pi}$ pulse. To second order in the pulse duration, we characterize a symmetric pulse shape by three parameters, two of which can be made zero by shaping. We express the effective Hamiltonians for several pulse sequences in terms of these parameters and use the results to analyze the structure of error operators for a controlled Jaynes-Cummings Hamiltonian. When errors are cancelled to second order, numerical simulations show excellent qubit fidelity with strongly suppressed oscillator heating.

arXiv:0708.4267Physical Review AJan 24, 200822 citations

Optimal arbitrarily accurate composite pulse sequences

Guang Hao Low, Theodore J. Yoder, Isaac L. Chuang

Implementing a single-qubit unitary is often hampered by imperfect control. Systematic amplitude errors $\ensuremath{\epsilon}$, caused by incorrect duration or strength of a pulse, are an especially common problem. But a sequence of imperfect pulses can provide a better implementation of a desired operation, as compared to a single primitive pulse. We find optimal pulse sequences consisting of $L$ primitive $\ensuremath{\pi}$ or $2\ensuremath{\pi}$ rotations that suppress such errors to arbitrary order $O({\ensuremath{\epsilon}}^{n})$ on arbitrary initial states. Optimality is demonstrated by proving an $L=O(n)$ lower bound and saturating it with $L=2n$ solutions. Closed-form solutions for arbitrary rotation angles are given for $n=1,2,3,4$ . Perturbative solutions for any $n$ are proven for small angles, while arbitrary angle solutions are obtained by analytic continuation up to $n=12$ . The derivation proceeds by a novel algebraic and nonrecursive approach, in which finding amplitude error correcting sequences can be reduced to solving polynomial equations.Implementing a single-qubit unitary is often hampered by imperfect control. Systematic amplitude errors $\ensuremath{\epsilon}$, caused by incorrect duration or strength of a pulse, are an especially common problem. But a sequence of imperfect pulses can provide a better implementation of a desired operation, as compared to a single primitive pulse. We find optimal pulse sequences consisting of $L$ primitive $\ensuremath{\pi}$ or $2\ensuremath{\pi}$ rotations that suppress such errors to arbitrary order $O({\ensuremath{\epsilon}}^{n})$ on arbitrary initial states. Optimality is demonstrated by proving an $L=O(n)$ lower bound and saturating it with $L=2n$ solutions. Closed-form solutions for arbitrary rotation angles are given for $n=1,2,3,4$. Perturbative solutions for any $n$ are proven for small angles, while arbitrary angle solutions are obtained by analytic continuation up to $n=12$. The derivation proceeds by a novel algebraic and nonrecursive approach, in which finding amplitude error correcting sequences can be reduced to solving polynomial equations.

arXiv:1307.2211Physical Review AFeb 28, 201444 citations

L

primitive $\ensuremath{\pi}$ or $2\ensuremath{\pi}$ rotations that suppress such errors to arbitrary order $O({\ensuremath{\epsilon}}^{n})$ on arbitrary initial states. Optimality is demonstrated by proving an

L=O(n)

lower bound and saturating it with

L=2n

solutions. Closed-form solutions for arbitrary rotation angles are given for

n=1,2,3,4

. Perturbative solutions for any

n

are proven for small angles, while arbitrary angle solutions are obtained by analytic continuation up to

n=12

. The derivation proceeds by a novel algebraic and nonrecursive approach, in which finding amplitude error correcting sequences can be reduced to solving polynomial equations.