Gate synthesis designs the controls that make a quantum device implement a target unitary — a single-qubit rotation, an entangling two-qubit gate, a multi-qubit or qudit operation — with an error small enough for computation. The fault-tolerance threshold of the surface code is about 1% error per operation [1], and practical overheads need errors well below it. Gate synthesis is the most common task in quantum optimal control and the main driver of its methods.

The problem

  • Objective. The standard figure of merit compares the achieved evolution with the target, ignoring a global phase, \(F = |\mathrm{Tr}(U_\mathrm{target}^\dagger U(T))|^2/d^2\) [2]. Average gate fidelities follow from it [3, 4]. When the device has more levels than the computational space, \(U(T)\) is projected onto that space, so population lost from it lowers \(F\). With dissipation, the objective is defined on the dynamical map [5]. See Problem formulation & figures of merit.
  • Targets up to local operations. For entangling gates, often only the nonlocal part matters. Two-qubit gates up to single-qubit operations form a tetrahedron (the Weyl chamber), and exactly half of the nonlocal gates are perfect entanglers [6]. One can optimize for any perfect entangler rather than a specific gate [7].
  • Existence and time. Controllability guarantees that a solution exists for long enough times. Speed limits and Geometric & time-optimal control set how short it can be. Building blocks of quantum algorithms can be synthesized directly, which can be faster than a decomposition into elementary gates [8]. Globally optimal solutions can be certified with QCPOP & polynomial optimization [9].

The errors control must handle

  • Leakage. Weakly anharmonic qubits such as transmons leak into higher levels when driven fast. DRAG removes leakage and phase errors to leading order [10]. Leakage can be measured and suppressed [11], and numerically optimized pulses reduce it further in fast gates [12]. Virtual \(Z\) gates, implemented as phase changes of later drives, correct rotation errors at no cost in time [13]. See DRAG & leakage suppression.
  • Crosstalk. Crosstalk covers many physical effects that make an operation on some qubits depend on others. It can be defined rigorously and detected operationally [14]. Control can address it: analytic pulses drive simultaneous gates in frequency-crowded systems [15], an extra drive tone cancels unwanted terms of the cross-resonance interaction [16], and coupler controls designed beyond the dispersive approximation remove leakage to the coupler and parasitic \(ZZ\) terms [17].
  • Decoherence. Shorter gates lose less coherence, which is why time-optimal solutions matter. Open-system optimization takes the loss into account directly; see Control of open quantum systems.
  • Model errors. Pulses optimized on an imperfect model are refined on the device, with randomized benchmarking as the cost [18] or by hybrid model-based and closed-loop schemes [19]. See Closed-loop & in-situ optimization and Robust & ensemble optimal control.

Entangling gates on different platforms

  • Superconducting qubits. The cross-resonance gate entangles fixed-frequency qubits with microwaves alone [20]. Its fidelity rose above 99% once its full Hamiltonian was calibrated and the unwanted terms cancelled [16]. Alternatives include parametrically activated gates [21] and tunable couplers, which reached 99.76% for CZ and 99.87% for iSWAP [17]. See Superconducting circuits.
  • Trapped ions. The Mølmer–Sørensen gate works even with ions in thermal motion [22]. In long chains, many motional modes take part, and optimally shaped, segmented pulses control the coupling to all of them [23]. Laser-driven gates have reached 99.9% for two qubits and single-qubit errors of a few \(10^{-5}\) [24, 25]. See Trapped ions.
  • Neutral atoms. The Rydberg blockade gives fast entangling gates [26]. A CZ gate driven by a global laser pulse avoids single-site addressing [27]. Optimal control found time-optimal CZ and CCZ pulses with only a few parameters [28]. Single-pulse gates of this kind reached 99.5% fidelity on up to 60 atoms in parallel [29]. See Neutral atoms & Rydberg arrays.
  • Qudits. Using more than two levels per carrier, a trapped-ion processor achieved universal control of qudits with up to seven levels [30].

Measuring success

Gate errors near \(10^{-4}\) cannot be resolved by process tomography, which is itself limited by state-preparation and measurement errors. Randomized benchmarking measures the average error of long random sequences instead [31, 32]. Gate set tomography gives a self-consistent model of all the operations [33]. See System identification for control.

Limitations

  • An isolated gate’s fidelity is not its performance in a circuit. Crosstalk, spectator qubits, drift and non-Markovian noise appear only in context.
  • Pulses that exploit fine details of a model can be fragile, so robustness has to be traded against speed.
  • Calibration drifts, and keeping many optimized pulses calibrated on a large device is an engineering problem in its own right. See Pulse-level compilation.