Noise is ubiquitous in quantum systems and is a major obstacle for the advancement of quantum information science. Noise-robust quantum control achieves high-fidelity operations by engineering the evolution path so that first-order noise contributions cancel at the final time. Such dynamical error correction typically incurs a time overhead beyond standard quantum speed limits. We derive general lower bounds on control complexity that quantify this overhead for quasi-static coherent noise under bounded control amplitude. For a single noise source, we prove a universal time lower bound for first-order robustness and give a constructive scheme that implements any target gate robustly in time 4T plus a constant time. For robustness against an entire noise space, we show dimension lower bounds on the number M of segments in any mixed-unitary schedule from two mechanism: (i) a coherent dimension bound when the error subspace contains an irreducible block isomorphic to su(q), and (ii) a projection dimension bound when the noise space contains the trace-zero span of orthogonal projectors. Under bounded speed, these bounds on number of segments imply time lower bounds. With only local controls robust against noise space defined on a graph, we obtain a graph-orthogonality time bound scales linear with graph chromatic number. We illustrate the bounds through examples. Collectively, these results establish quantitative limitations on the feasibility of first-order noise-resilient operations.
Noise is ubiquitous in quantum systems and is a major obstacle for the advancement of quantum information science. Noise-robust quantum control achieves high-fidelity operations by engineering the evolution path so that first-order noise contributions cancel at the final time. Such dynamical error correction typically incurs a time overhead beyond standard quantum speed limits. We derive general lower bounds on control complexity that quantify this overhead for quasi-static coherent noise under bounded control amplitude. For a single noise source, we prove a universal time lower bound for first-order robustness and give a constructive scheme that implements any target gate robustly in time 4T plus a constant time. For robustness against an entire noise space, we show dimension lower bounds on the number M of segments in any mixed-unitary schedule from two mechanism: (i) a coherent dimension bound when the error subspace contains an irreducible block isomorphic to su(q), and (ii) a projection dimension bound when the noise space contains the trace-zero span of orthogonal projectors. Under bounded speed, these bounds on number of segments imply time lower bounds. With only local controls robust against noise space defined on a graph, we obtain a graph-orthogonality time bound scales linear with graph chromatic number. We illustrate the bounds through examples. Collectively, these results establish quantitative limitations on the feasibility of first-order noise-resilient operations.
We derive a universal performance limit for coherent quantum control in the presence of modeled and unmodeled uncertainties. For any target unitary W that is implementable in the absence of error, we prove that the worst-case (and hence the average) gate fidelity obeys the lower bound $F \ge \Flb\bigl(\tf \Omeff\bigr)$, where $\tf$ is the gate duration and $\Omeff$ is a single frequency-like measure that aggregates \emph{all} bounded uncertainty sources, e.g., coherent control imperfections, unknown couplings, and residual environment interactions, without assuming an initially factorizable system-bath state or a completely positive map. The bound is obtained by combining an interaction-picture averaging method with a Bellman-Gronwall inequality and holds for any finite-norm Hamiltonian decomposition. Hence it applies equally to qubits, multi-level qudits, and ancilla-assisted operations. Because $\Flb$ depends only on the dimensionless product $\tf\Omeff$, it yields a device-independent metric that certifies whether a given hardware platform can, in principle, reach a specified fault-tolerance threshold, and also sets a quantitative target for robust-control synthesis and system identification.
Time and space (memory) are two of the most important measures of cost in computation, even more so for quantum computation. Yet, our tools for proving unconditional quantum tradeoffs between time and space are surprisingly limited.
The first quantum time-space tradeoff lower bounds were proven for sorting by Klauck, Špalek and de Wolf. Unfortunately, their method is limited to proving output-oblivious lower bounds (i.e. the lower bounds only apply to algorithms with a non-adaptive output schedule) and other methods have yielded nothing beyond output-oblivious lower bounds for sorting. Here, we prove the first fully general quantum time-space tradeoff lower bound for sorting.
We do so by introducing a novel method based on the noise operator to add to the analysis toolkit for proving quantum query and time-space tradeoff lower bounds. By combining our resulting quantum noise stability bound with quantum recording query methods, we prove an Ω(n4/3(loglogn)/(S1/3logn)) lower bound on the number of queries that a fully general quantum algorithm with at most S qubits of memory requires to sort n numbers from [n2]. Applying our noise operator argument involves purely classical reasoning, which makes it particularly simple to use.
We also use it to prove that, for any strongly universal (pairwise independent) hash function family H from n bits to m bits, almost all hash functions in H require any algorithm with at most S qubits of memory to make Ω(nm/S) quantum queries to input x in order to compute h(x), even with very small success probability. Previously, Mansour, Nisan, and Tiwari had shown a matching classical lower bound for computing h(x) with both h and x as inputs using their hash mixing lemma. Our noise operator method allows us to use a related property of hash functions to prove our quantum lower bounds.
We extend the work in 2017 New J. Phys. 19 103015 by deriving a lower bound for the minimum time necessary to implement a unitary transformation on a generic, closed quantum system with an arbitrary number of classical control fields. This bound is explicitly analyzed for a specific N -level system similar to those used to represent simple models of an atom, or the first excitation sector of a Heisenberg spin chain, both of which are of interest in quantum control for quantum computation. Specifically, it is shown that the resultant bound depends on the dimension of the system, and on the number of controls used to implement a specific target unitary operation. The value of the bound determined numerically, and an estimate of the true minimum gate time are systematically compared for a range of system dimension and number of controls; special attention is drawn to the relationship between these two variables. It is seen that the bound captures the scaling of the minimum time well for the systems studied, and quantitatively is correct in the order of magnitude.
Robust quantum control usually assumes one of two extremes: the error is of a known form and only its strength is unknown, or nothing at all is known about it. In practice one often sits in between. The size of a static error is unknown, but its class is known, for example, it only shifts energy levels, or it acts on each subsystem separately. We ask how fast a control can cancel, to first order, every error in such a class. A time-independent Hamiltonian can do this if and only if all errors in the class have zero diagonal in a common basis. In that case an equally spaced energy ladder in this basis does it after one period. A lower bound that holds for any time-dependent control with bounded drive power shows that this simple constant control is optimal for several natural classes, including unknown level shifts and unknown local errors on two subsystems. Completely unknown errors need a time-dependent control, and there, knowing the error class shortens the minimum time by a factor d2+1. So partial knowledge of the error changes not only how long an optimal robust control takes, but also its form.
Quantum computers can significantly speed up the solution of certain classical tasks and even provide solutions to problems that are classically intractable. However, achieving this requires a fault-tolerant architecture with components and quantum operations (gates) whose physical error rates are below a specific threshold. Advanced quantum control methods can suppress environmental noise and implement below-threshold gates, but there are infinitely many ways to do so. These methods can rely entirely on analytical solutions or combine them with numerical optimization, where pulse properties are treated as optimization goals. Analytical constraints are always met exactly, but numerically optimized goals may only be satisfied approximately, leading to tradeoffs. Therefore, a modular analytical framework is needed that fixes some objectives in advance while optimizing only a selected subset. In this dissertation, I employ Space Curve Quantum Control (SCQC), a method that generates noise-resistant quantum gates using differential geometry. In SCQC, a space curve's global geometric properties dictate the gate's noise-robustness, while the controls that realize it are extracted from its local quantities. The first part of the dissertation is devoted to extending SCQC for arbitrary and simultaneous noise suppression, with experimentally-friendly pulses. I first derive the geometric conditions and curve examples for suppressing two separate noise types simultaneously. Then, I extend SCQC by deriving the adjoint representation equations of the associated quantum evolution. This allows one to describe the entire control design in terms of space curve elements and propose a systematic control design approach called the Bézier Ansatz for Robust Quantum (BARQ) control method. BARQ can encode the target gate upfront and allow optimization only for noise-robustness properties. This approach solves the usually unavoidable tradeoff between gate fidelity and noise-sensitivity and is capable of producing control pulses with experimentally-friendly features. The second part of the dissertation applies BARQ in two leading qubit modalities. I first describe my participation in experiments on IBM's superconducting devices, where doubly-robust BARQ pulses demonstrate at least one order-of-magnitude error reduction in the presence of engineered noise. The second platform is a four-qubit trapped-ion register hosted in the Quantum Scientific Computing Open User Testbed (QSCOUT) ecosystem. I show that utilizing BARQ pulses can reduce gate errors up to two times and effectively combat motion-induced decoherence.
Resisting diverse noise effects is crucial for the accurate manipulation of quantum systems. For static or slowly varying noises, many efficient noise mitigation strategies have been developed, such as composite pulses and dynamical decoupling. However, for fast fluctuating noise in the Markovian limit, whether and to what extent coherent quantum control can enhance quantum engineering tasks remains unclear and less explored. Here, we propose a robust quantum optimal control method to tackle Markovian noises. The basic idea is that, regarding the Markovian noise channel as perturbation, we quantitatively characterize the noise-induced error evolution by the perturbative expansion term, and then take them as the objective functions to be suppressed. During optimization, the optimal controls are obtained by maximizing the control target function and meanwhile minimizing the perturbative terms due to Markovian noise order by order. As demonstration examples, we first apply our method to quantum state transfer tasks on two-level and three-level $\mathrm{\ensuremath{\Lambda}}$ systems, then use it to design quantum gates in two-level and three-level ladder systems, all under Markovian noises. The simulation results illustrate that our method can notably enhance quantum state transfer fidelities and have very limited improvement on gate fidelities. The method presented here is versatile and can be extended to enhance the performance of various control tasks under Markovian noise in multilevel or multiqubit quantum systems.