There are well-known protocols for performing controlled NOT (CNOT) quantum logic with qubits coupled by particular high-symmetry (Ising or Heisenberg) interactions. However, many architectures being considered for quantum computation involve qubits or qubits and resonators coupled by more complicated and less symmetric interactions. Here we consider a widely applicable model of weakly but otherwise arbitrarily coupled two-level systems, and use quantum gate design techniques to derive a simple and intuitive CNOT construction. Useful variations and extensions of the solution are given for common special cases.
This work studies the feasibility of optimal control of high-fidelity quantum gates in a model of interacting two-level particles. One particle (the qubit) serves as the quantum information processor, whose evolution is controlled by a time-dependent external field. The other particles are not directly controlled and serve as an effective environment, coupling to which is the source of decoherence. The control objective is to generate target one-qubit gates in the presence of strong environmentally-induced decoherence and under physically motivated restrictions on the control field. It is found that interactions among the environmental particles have a negligible effect on the gate fidelity and require no additional adjustment of the control field. Another interesting result is that optimally controlled quantum gates are remarkably robust to random variations in qubit-environment and inter-environment coupling strengths. These findings demonstrate the utility of optimal control for management of quantum-information systems in a very precise and specific manner, especially when the dynamics complexity is exacerbated by inherently uncertain environmental coupling.
A possible building block for a scalable quantum computer has recently been demonstrated [Mariantoni et al., Science 334, 61 (2011)]. This architecture consists of superconducting qubits capacitively coupled both to individual memory resonators as well as a common bus. In this work we study a natural primitive entangling gate for this and related resonator-based architectures, which consists of a controlled-${\ensuremath{\sigma}}^{z}$ (cz) operation between a qubit and the bus. The cz gate is implemented with the aid of the noncomputational qubit $|2\ensuremath{\rangle}$ state [Strauch et al., Phys. Rev. Lett. 91, 167005 (2003)]. Assuming phase or transmon qubits with 300 MHz anharmonicity, we show that by using only low frequency qubit-bias control it is possible to implement the qubit-bus cz gate with 99.9
(99.99
) fidelity in about
17ns (
23ns) with a realistic two-parameter pulse profile, plus two auxiliary
z rotations. The fidelity measure we refer to here is a state-averaged intrinsic process fidelity, which does not include any effects of noise or decoherence. These results apply to a multiqubit device that includes strongly coupled memory resonators. We investigate the performance of the qubit-bus cz gate as a function of qubit anharmonicity, identify the dominant intrinsic error mechanism and derive an associated fidelity estimator, quantify the pulse shape sensitivity and precision requirements, simulate qubit-qubit cz gates that are mediated by the bus resonator, and also attempt a global optimization of system parameters including resonator frequencies and couplings. Our results are relevant for a wide range of superconducting hardware designs that incorporate resonators and suggest that it should be possible to demonstrate a
99.9 cz gate with existing transmon qubits, which would constitute an important step towards the development of an error-corrected superconducting quantum computer.
Building on the previous results of the Weyl chamber steering method, we demonstrate how to generate high-fidelity controlled-NOT (CNOT) gates by direct application of certain physically relevant Hamiltonians with fixed coupling constants containing Rabi terms. Such Hamiltonians are often used to describe two superconducting qubits driven by local rf pulses. It is found that in order to achieve 100% fidelity in a system with capacitive coupling of strength
g, one Rabi term suffices. We give the exact values of the physical parameters needed to implement such CNOT gates. The gate time and all possible Rabi frequencies are found to be $t=\ensuremath{\pi}∕(2g)$ and ${\ensuremath{\Omega}}_{1}∕g=\sqrt{64{n}^{2}\ensuremath{-}1},n=1,2,3,\dots{}$. Generation of a perfect CNOT gate in a system with inductive coupling, characterized by additional constant
k, requires the presence of both Rabi terms. The gate time is again $t=\ensuremath{\pi}∕(2g)$, but now there is an infinite number of solutions, each of which is valid in a certain range of
k and is characterized by a pair of integers
(n,m), $\frac{{\ensuremath{\Omega}}_{1,2}}{g}=\sqrt{16{n}^{2}\ensuremath{-}{(\frac{k\ensuremath{-}1}{2})}^{2}}\ifmmode\pm\else\textpm\fi{}\sqrt{16{m}^{2}\ensuremath{-}{(\frac{k+1}{2})}^{2}}$. We distinguish two cases, depending on the sign of the coupling constant: (i) the antiferromagnetic case $(k\ensuremath{\geqslant}0)$ with $n\ensuremath{\geqslant}m=0,1,2,\dots{}$ and (ii) the ferromagnetic case $(k\ensuremath{\leqslant}0)$ with
n>m=0,1,2,…. We conclude with consideration of fidelity degradation by switching to resonance. Simulation of time evolution based on the fourth-order Magnus expansion reveals characteristics of the gate similar to those found in the exact case, with slightly shorter gate time and shifted values of the Rabi frequencies.
The fidelity of quantum operations is often limited by incoherent errors, which typically can be modeled by fundamental Markovian noise processes such as amplitude damping and dephasing. In Phys. Rev. Lett. 129, 150504 (2022), we presented an analytical result for the average gate fidelity of a general multiqubit operation in terms of the dissipative rates and the corresponding Lindblad jump operators, provided that the operation remains in the computational subspace throughout the time evolution. Here we generalize this expression for the average gate fidelity to include the cases where the system state temporarily leaves the computational subspace during the gate. Such gate mechanisms are integral to several quantum-computing platforms, and our formula is applicable to all of them; as examples, we employ it for the two-qubit controlled-Z gate in both superconducting qubits and neutral atoms. We also obtain the average gate fidelity for simultaneous operations applied in multiqubit systems. These results are useful for understanding the error budgets of quantum gates while scaling up quantum computers.
One of the largest obstacles to building a quantum computer is gate error, where the physical evolution of the state of a qubit or group of qubits during a gate operation does not match the intended unitary transformation. Gate error stems from a combination of control errors and random single qubit errors from interaction with the environment. While great strides have been made in mitigating control errors, intrinsic qubit error remains a serious problem that limits gate fidelity in modern qubit architectures. Simultaneously, recent developments of small error-corrected logical qubit devices promise significant increases in logical state lifetime, but translating those improvements into increases in gate fidelity is a complex challenge. In this Letter, we construct protocols for gates on and between small logical qubit devices which inherit the parent device's tolerance to single qubit errors which occur at any time before or during the gate. We consider two such devices, a passive implementation of the three-qubit bit flip code, and the author's own [E. Kapit, Phys. Rev. Lett. 116, 150501 (2016)PRLTAO0031-900710.1103/PhysRevLett.116.150501] very small logical qubit (VSLQ) design, and propose error-tolerant gate sets for both. The effective logical gate error rate in these models displays superlinear error reduction with linear increases in single qubit lifetime, proving that passive error correction is capable of increasing gate fidelity. Using a standard phenomenological noise model for superconducting qubits, we demonstrate a realistic, universal one- and two-qubit gate set for the VSLQ, with error rates an order of magnitude lower than those for same-duration operations on single qubits or pairs of qubits. These developments further suggest that incorporating small logical qubits into a measurement based code could substantially improve code performance.