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arXiv:1201.3360 · New Journal of Physics

Selective darkening of degenerate transitions for implementing quantum controlled-NOT gates

P. C. de Groot, Sahel Ashhab, Adrian Lupaşcu, L. DiCarlo, Franco Nori, C. J. P. M. Harmans, Johan E. Mooij

Published Jul 17, 2012• 30 citations
arXivHTMLPDF

Abstract

We present a theoretical analysis of the selective darkening method for implementing quantum controlled-NOT (CNOT) gates. This method, which we have recently proposed and demonstrated, consists of driving two transversely coupled quantum bits (qubits) with a driving field that is resonant with one of the two qubits. For specific relative amplitudes and phases of the driving field felt by the two qubits, one of the two transitions in the degenerate pair is darkened or, in other words, becomes forbidden by effective selection rules. In these driving conditions, the evolution of the two-qubit state realizes a CNOT gate. The gate speed is found to be limited only by the coupling energy J , which is the fundamental speed limit for any entangling gate. Numerical simulations show that at gate speeds corresponding to 0.48 J and 0.07 J , the gate fidelity is 99% and 99.99%, respectively, and increases further for lower gate speeds. In addition, the effect of higher-lying energy levels and weak anharmonicity is studied, as well as the scalability of the method to systems of multiple qubits. We conclude that in all these respects this method is competitive with existing schemes for creating entanglement, with the added advantages of being applicable for qubits operating at fixed frequencies (either by design or for the exploitation of coherence sweet-spots) and having the simplicity of microwave-only operation.

DOI
10.1088/1367-2630/14/7/073038

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We present a theoretical analysis of the selective darkening method for implementing quantum controlled-NOT (CNOT) gates. This method, which we have recently proposed and demonstrated, consists of driving two transversely coupled quantum bits (qubits) with a driving field that is resonant with one of the two qubits. For specific relative amplitudes and phases of the driving field felt by the two qubits, one of the two transitions in the degenerate pair is darkened or, in other words, becomes forbidden by effective selection rules. In these driving conditions, the evolution of the two-qubit state realizes a CNOT gate. The gate speed is found to be limited only by the coupling energy J , which is the fundamental speed limit for any entangling gate. Numerical simulations show that at gate speeds corresponding to 0.48 J and 0.07 J , the gate fidelity is 99% and 99.99%, respectively, and increases further for lower gate speeds. In addition, the effect of higher-lying energy levels and weak anharmonicity is studied, as well as the scalability of the method to systems of multiple qubits. We conclude that in all these respects this method is competitive with existing schemes for creating entanglement, with the added advantages of being applicable for qubits operating at fixed frequencies (either by design or for the exploitation of coherence sweet-spots) and having the simplicity of microwave-only operation.
We analyze the cross-resonance effect for fluxonium circuits and investigate a two-qubit gate scheme based on selective darkening of a transition. In this approach, two microwave pulses at the frequency of the target qubit are applied simultaneously with a proper ratio between their amplitudes to achieve a controlled-not operation. We study in detail coherent gate dynamics and calculate gate error. With nonunitary effects accounted for, we demonstrate that gate error below ${10}^{\ensuremath{-}4}$ is possible for realistic hardware parameters. This number is facilitated by long coherence times of computational transitions and strong anharmonicity of fluxoniums, which easily prevents excitation to higher excited states during the gate microwave drive.
We consider the implementation of two-qubit gates when the physical systems used to realize the qubits are weakly anharmonic and therefore possess additional quantum states in the accessible energy range. We analyze the effect of the additional quantum states on the maximum achievable speed for quantum gates in the qubit state space. By calculating the minimum gate time using optimal control theory, we find that higher energy levels can help make two-qubit gates significantly faster than the reference value based on simple qubits. This speedup is a result of the higher coupling strength between higher energy levels. We then analyze the situation where the pulse optimization algorithm avoids pulses that excite the higher levels. We find that in this case the presence of the additional states can lead to a significant reduction in the maximum achievable gate speed. We also compare the optimal control gate times with those obtained using the cross-resonance/selective-darkening gate protocol. We find that the latter, with some parameter optimization, can be used to achieve a relatively fast implementation of the CNOT gate. These results can help the search for optimized gate implementations in realistic quantum computing architectures, such as those based on superconducting qubits. They also provide guidelines for desirable conditions on anharmonicity that would allow optimal utilization of the higher levels to achieve fast quantum gates.
Controlled-NOT (CNOT) gates are commonly included in the standard gate set of quantum processors and provide an important way to entangle qubits. For fixed-frequency qubits using the cross-resonance entangling technique, using the higher-frequency qubit to control the lower-frequency qubit enables much shorter entangling times than using the lower-frequency qubit as the control. Consequently, when implementing a CNOT gate where logical control by the lower-frequency qubit is needed, compilers may implement this functionality by using an equivalent circuit such as placing Hadamard gates on both qubits before and after a CNOT gate controlled by the higher-frequency qubit. However, since the implementation is different depending on which qubit is the control, a natural question arises regarding the relative performance of the implementations. We have explored this using quantum processors on the IBM Q network. The basic circuit used consisted of operations to create a Bell State, followed by the inverse operations so as to return the qubits to their initial state in the absence of errors (Hadamard + CNOT + barrier + CNOT + Hadamard). The circuit depth was varied using multiples of this basic circuit. An asymmetry in the error of the final state was observed that increased with the circuit depth. The strength and direction of the asymmetry was unique but repeatable for each pair of coupled qubits tested. This observation suggests that the asymmetry in CNOT implementation should be characterized for the qubits of interest and incorporated into circuit transpilation to obtain the best accuracy for a particular computation.
We establish theoretical bounds on qubit detuning for some of the previously proposed controlled-NOT (CNOT) logic gate implementations with weakly coupled Josephson phase qubits. It is found that in the two-step, iSWAP\sqrt{i\text{SWAP}}iSWAP​-based case the value of the detuning during the entangling operations must not exceed 2g2g2g, where ggg is the characteristic coupling constant. In the single-step case we consider two practical, physically distinct implementations, in which one of the qubits is driven by a concurrent rf pulse of fixed frequency. We find that when the local drive is applied to the ``reference'' qubit (with which it is in resonance), the detuning should not exceed ggg. If the drive is applied to the ``detuned'' qubit, generation of the perfect CNOT gate is possible at any value of detuning provided that the amplitude of the pulse can be made arbitrarily large.
We consider the implementation of two-qubit gates when the physical systems used to realize the qubits possess additional quantum states in the accessible energy range. We use optimal control theory to determine the maximum achievable gate speed for two-qubit gates in the qubit subspace of the many-level Hilbert space, and we analyze the effect of the additional quantum states on the gate speed. We identify two competing mechanisms. On one hand, higher energy levels are generally more strongly coupled to each other. Under suitable conditions, this stronger coupling can be utilized to make two-qubit gates significantly faster than the reference value based on simple qubits. On the other hand, a weak anharmonicity constrains the speed at which the system can be adequately controlled: according to the intuitive picture, faster operations require stronger control fields, which are more likely to excite higher levels in a weakly anharmonic system, which in turn leads to faster decoherence and uncontrolled leakage outside the qubit space. To account for this constraint, we modify the pulse optimization algorithm to avoid pulses that lead to appreciable population of the higher levels. In this case, we find that the presence of the higher levels can lead to a significant reduction in the maximum achievable gate speed. We also compare the optimal-control gate speeds with those obtained using the cross-resonance or selective-darkening gate protocol. We find that this protocol, with some parameter optimization, can be used to achieve a relatively fast implementation of the controlled-NOT gate. These results can help the search for optimized gate implementations in realistic quantum computing architectures, such as those based on superconducting circuits. They also provide guidelines for desirable conditions on anharmonicity that would allow optimal utilization of the higher levels to achieve fast quantum gates.
We model a controlled-NOT (CNOT) gate mediated by exchange interactions in a realistic Ge/GeSi hole spin-qubit device. We use for that purpose a computationally efficient framework based on a compact dressed basis of many-body singlet and triplet states, which enables fast time-dependent simulations that would be prohibitive within a conventional configuration-interaction approach. We discuss the dominant sources of infidelity in ideal (noise-free) devices. On the one hand, the nonadiabaticity of the tunnel barrier ramps, enhanced by the transverse component of the exchange interaction, imposes limits on the speed of the CNOT gate. On the other hand, the AC Stark shift of the conditional resonance and the off-resonant driving of the complementary transition can significantly degrade the gate fidelity and call for a careful optimization of the driving conditions. Finally, we investigate the impact of quasi-static charge noise on the fidelity of the CNOT gate. We find that the capture of charges near the qubits can strongly disrupt the conditional operation, while the nanometer-scale motion of charges results in a loss of fidelity that sensitively depends on the position and strength of the corresponding dipole. Our results quantify the requirements on barrier control, microwave driving, and charge-noise stability for high-fidelity two-qubit operations in Ge/GeSi hole-spin qubits.