We use analytical and numerical calculations to obtain speed limits for various unitary quantum operations in multiqubit systems under typical experimental conditions. The operations that we consider include single-, two-, and three-qubit gates, as well as quantum-state transfer in a chain of qubits. We find in particular that simple methods for implementing two-qubit gates generally provide the fastest possible implementations of these gates. We also find that the three-qubit Toffoli gate time varies greatly depending on the type of interactions and the system's geometry, taking only slightly longer than a two-qubit controlled-not (cnot) gate for a triangle geometry. The speed limit for quantum-state transfer across a qubit chain is set by the maximum spin-wave speed in the chain.
We use analytical and numerical calculations to obtain speed limits for various unitary quantum operations in multiqubit systems under typical experimental conditions. The operations that we consider include single-, two-, and three-qubit gates, as well as quantum-state transfer in a chain of qubits. We find in particular that simple methods for implementing two-qubit gates generally provide the fastest possible implementations of these gates. We also find that the three-qubit Toffoli gate time varies greatly depending on the type of interactions and the system's geometry, taking only slightly longer than a two-qubit controlled-not (cnot) gate for a triangle geometry. The speed limit for quantum-state transfer across a qubit chain is set by the maximum spin-wave speed in the chain.
The speed of elementary quantum gates, particularly two-qubit gates, ultimately sets the limit on the speed at which quantum circuits can operate. In this work, we experimentally demonstrate commonly used two-qubit gates at nearly the fastest possible speed allowed by the physical interaction strength between two superconducting transmon qubits. We achieve this quantum speed limit by implementing experimental gates designed using a machine-learning-inspired optimal control method. Importantly, our method only requires the single-qubit drive strength to be moderately larger than the interaction strength to achieve an arbitrary two-qubit gate close to its analytical speed limit with high fidelity. Thus the method is applicable to a variety of platforms, including those with comparable single-qubit and two-qubit gate speeds, or those with always-on interactions. We expect our method to offer significant speedups for non-native two-qubit gates that are typically achieved with a long sequence of single-qubit and native two-qubit gates.
The speed of quantum evolution is limited under finite energy resources. While most quantum speed limits (QSLs) are formulated in terms of quantum states, they can be extended to the evolution operator itself, and thus impose fundamental limits on how quickly logical gate operations can be implemented on a quantum computer. Here, we derive a general, tight QSL that holds for any unitary evolution under the constraint that the spectral width of the Hamiltonian is bounded. We apply this result to obtain QSLs for several standard quantum gates, including Hadamard, CNOT, and Toffoli gates, finding that the QSL can vary significantly across different gates, including ones with the same entangling power. These findings can be understood geometrically using the Space Curve Quantum Control formalism, which maps unitary evolution to space curves in Euclidean space. In this formalism, the problem of finding QSLs is recast as the problem of finding minimal-length curves obeying a curvature bound. We find that time-optimal gates map to helices of varying dimensions, and that QSLs can be understood from the perspective of a bottleneck principle in which the operator that evolves the slowest governs the minimal gate time.
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.
The speed of elementary quantum gates ultimately sets the limit on the speed at which quantum circuits can operate. For a fixed physical interaction strength between two qubits, the speed of any two-qubit gate is limited even with arbitrarily fast single-qubit gates. In this work, we explore the possibilities of speeding up two-qubit gates beyond such a limit by expanding our computational space outside the qubit subspace, which is experimentally relevant for qubits encoded in multi-level atoms or anharmonic oscillators. We identify an optimal theoretical bound for the speed limit of a two-qubit gate achieved using two qudits with a bounded interaction strength and arbitrarily fast single-qudit gates. In addition, we find an experimentally feasible protocol using two parametrically coupled superconducting transmons that achieves this theoretical speed limit in a non-trivial way. We also consider practical scenarios with limited single-qudit drive strengths and off-resonant transitions. For such scenarios, we develop an open-source, machine learning assisted, quantum optimal control algorithm that can achieve a speedup close to the theoretical limit with near-perfect gate fidelity. This work opens up a new avenue to speed up two-qubit gates when the physical interaction strength between qubits cannot be easily increased while extra states outside the qubit subspace can be well controlled.
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.
Single-step implementations of multiqubit gates are generally believed to provide a simpler design, a faster operation, and a lower decoherence. For three coupled qubits interacting with a photon field, a realizable scheme for a single-step Toffoli gate is investigated. We find that the three-qubit system can be described by four effective modified Jaynes-Cummings models in the states of two control qubits. Within the rotating-wave approximation, the modified Jaynes-Cummings models are shown to be reduced to the conventional Jaynes-Cummings models with renormalized couplings between qubits and photon fields. A single-step Toffoli gate is shown to be realizable with tuning of the four characteristic oscillation periods that satisfy a commensurate condition. Possible values of the system parameters are estimated for a single-step Toffoli gate. Further, we discuss our single-step Toffoli-gate operation errors due to imperfections in the system parameters; numerical calculations show that a Toffoli gate with high fidelity can be obtained by adjusting pairs of the photon-qubit and qubit-qubit coupling strengths. In addition, a decoherence effect on the Toffoli-gate operation due to a thermal reservoir is discussed.