Successful implementations of quantum technologies require protocols and algorithms that use as few quantum resources as possible. However, many important quantum operations, such as continuous rotation gates in quantum computing or broadband pulses in NMR or magnetic resonance imaging (MRI) applications, can only be implemented approximately using finite quantum resources. This work develops an approach that, on average—at the cost of a modestly increased measurement repetition rate—enables exact implementations. One proceeds by first building a library of a large number of different approximations to the desired gate operation; by randomly selecting these operations according to a preoptimized probability distribution, one can on average implement the desired operation with a rigorously controllable approximation error. The approach relies on sophisticated tools from convex optimization to efficiently find optimal probability distributions. A diverse spectrum of applications are demonstrated as (a) exactly synthesizing rotations in fault-tolerant quantum computers using only low T -count circuits and (b) synthesizing broadband and band-selective pulses of superior performance in quantum optimal control with (c) further applications in NMR or MRI. The approach is very general and a broad spectrum of practical applications in quantum technologies are explicitly demonstrated. Published by the American Physical Society 2024
Successful implementations of quantum technologies require protocols and algorithms that use as few quantum resources as possible. However, many important quantum operations, such as continuous rotation gates in quantum computing or broadband pulses in NMR or magnetic resonance imaging (MRI) applications, can only be implemented approximately using finite quantum resources. This work develops an approach that, on average—at the cost of a modestly increased measurement repetition rate—enables exact implementations. One proceeds by first building a library of a large number of different approximations to the desired gate operation; by randomly selecting these operations according to a preoptimized probability distribution, one can on average implement the desired operation with a rigorously controllable approximation error. The approach relies on sophisticated tools from convex optimization to efficiently find optimal probability distributions. A diverse spectrum of applications are demonstrated as (a) exactly synthesizing rotations in fault-tolerant quantum computers using only low T -count circuits and (b) synthesizing broadband and band-selective pulses of superior performance in quantum optimal control with (c) further applications in NMR or MRI. The approach is very general and a broad spectrum of practical applications in quantum technologies are explicitly demonstrated. Published by the American Physical Society 2024
Preparing a quantum circuit that implements a given sparse state is an important building block that is necessary for many different quantum algorithms. In the context of fault-tolerant quantum computing, the so-called non-Clifford gates are much more expensive to perform than the Clifford ones. We hence provide an algorithm for synthesizing sparse quantum states with a special care for quantum resources. The circuit depth, ancilla count, and crucially non-Clifford count of the circuit produced by the algorithm are all linear in the sparsity when access to arbitrary-angled rotations is given. When compiled down to the standard Clifford+T gate set, several constructions can be given for increasingly better T-count and depth at the expense of a larger number of ancillae. The most optimised construction for T-count reaches O(s+log2(1/ε)) T gates for error ε. The constructions are broken into two parts, one that synthesises a generalized W-state, well studied in the literature; and the second which is a classical reversible circuit implementing a permutation that maps the basis states of the W-state to those of the target sparse quantum state. We reduce this problem to the diagonalization of a binary matrix, using a specific set of elementary matrix operations corresponding to the classical reversible gates. We then solve this problem using a new version of Gauss-Jordan elimination, that minimizes the circuit complexities including circuit depth using parallel elimination steps. When the circuit is applied in one direction, we notice that all occurrences of (the expensive) Toffoli gates can all be replaced by adaptive Clifford circuits, leading to a better non-Clifford count.
Rotation operations are essential ingredients in quantum algorithms. However, fault-tolerant quantum computation is intrinsically digital, based on a finite set of primitive gates, and thus cannot provide an analog rotation operation with arbitrary angles. It must resort to approximating a rotation operation using a basic universal gate set. Prior work on rotation synthesis either focuses on synthesizing individual rotations without considering a quantum circuit as a whole or is restricted to the specific quantum Fourier transform circuit. In this work, we develop a holistic and general approach to rotation synthesis, taking all rotation operations of a circuit into account for quantum cost minimization. Experimental results demonstrate the superiority of our method compared to the state-of-the-art methods in reducing the cost of synthesized Quantum algorithms with rotation operations.
Clifford+T synthesis of rotation gates is an important routine in fault-tolerant quantum compilation. While Clifford+T synthesis is scalable, it has a high overhead of tens of T gates per rotation in practice, translating to high resource estimates for many fault-tolerant algorithms. However, these well-known results, including those using probabilistic mixtures [Quantum 7, 1208 (2023)], are independent of the rotation angle θ, requiring O(log1/δ) T gates. We show that it is possible to do much better for small angles, reducing the T cost to O~(θ2/δ), and returning to existing O(log1/δ) results in the worst case. This is particularly important since many algorithms, such as Trotterization, are dominated by small-angle rotations. Further, we perform a detailed theoretical and numerical study of quasi-probabilities, which can further reduce the total T cost of large circuits by orders of magnitude with only a small overhead in sample complexity. We also develop a scheme based on quasi-probability mixtures of Clifford+T fallback channels. We derive new θ-dependent formulas that can be used for resource estimation of fault-tolerant quantum algorithms. As an application of our results, we show that the gate cost of Trotterization circuits compiled to a Clifford+T gate set is constant in the small Trotter step size limit, and can be reduced by orders of magnitude even for large step sizes. The cost of fault-tolerant Trotterization for a variety of applications should be re-examined in light of these results. Our work dispels the widely-stated claim that Clifford+T rotation synthesis has a high cost independent of θ, and further develops a scalable quasi-probability method for rotation synthesis. We also expect our results to bring forward useful early fault-tolerant quantum computing by reducing required magic state resources.
Quantum computing requires a universal set of gate operations; regarding gates as rotations, any rotation angle must be possible. However a real device may only be capable of B bits of resolution, i.e., it might support only 2^{B} possible variants of a given physical gate. Naive discretization of an algorithm's gates to the nearest available options causes coherent errors, while decomposing an impermissible gate into several allowed operations increases circuit depth. Conversely, demanding higher B can greatly complexify hardware. Here, we explore an alternative: probabilistic angle interpolation (PAI). This effectively implements any desired, continuously parametrized rotation by randomly choosing one of three discretized gate settings and postprocessing individual circuit outputs. The approach is particularly relevant for near-term applications where one would in any case average over many runs of circuit executions to estimate expected values. While PAI increases that sampling cost, we prove that (a) the approach is optimal in the sense that PAI achieves the least possible overhead and (b) the overhead is remarkably modest even with thousands of parametrized gates and only seven bits of resolution available. This is a profound relaxation of engineering requirements for first generation quantum computers where even 5-6 bits of resolution may suffice and, as we demonstrate, the approach is many orders of magnitude more efficient than prior techniques. Moreover we conclude that, even for more mature late noisy intermediate-scale quantum era hardware, no more than nine bits will be necessary.
Randomized protocols are procedures that incorporate probabilistic choices during their execution and they play a central role in quantum algorithms, spanning Hamiltonian simulation, noise mitigation, and measurement tasks. In practical implementations, the dominant cost of such protocols typically arises from circuit execution and measurement, and depends on hardware-specific resources such as gate counts, circuit depth, runtime, or dissipated energy. We introduce a general framework for applying classical importance sampling to randomized quantum protocols. Given a cost function for running quantum circuits, the proposed approach minimizes a net-cost figure of merit that jointly captures the computational expense per circuit and the estimator variance. We further extend the framework to scenarios where the quantum computation is subject to errors arising either from algorithmic approximations or from physical noise, proving that importance sampling preserves estimator bias despite altering the sampling distribution, and to settings with error-detection schemes, where we characterize the resulting changes in the optimal sampling strategy and achievable net-cost reduction. Representative applications include the Qdrift protocol, dephasing channels, mixed-states simulation, composite observables estimation, classical shadows, and probabilistic error cancellation. Overall, our results establish a principled approach for reducing the computational resources required by randomized quantum protocols through classical sampling optimization.
Quantum circuit synthesis is the process in which an arbitrary unitary operation is decomposed into a sequence of gates from a universal set, typically one which a quantum computer can implement both efficiently and fault-tolerantly. As physical implementations of quantum computers improve, the need is growing for tools that can effectively synthesize components of the circuits and algorithms they will run. Existing algorithms for exact, multi-qubit circuit synthesis scale exponentially in the number of qubits and circuit depth, leaving synthesis intractable for circuits on more than a handful of qubits. Even modest improvements in circuit synthesis procedures may lead to significant advances, pushing forward the boundaries of not only the size of solvable circuit synthesis problems, but also in what can be realized physically as a result of having more efficient circuits. We present a method for quantum circuit synthesis using deterministic walks. Also termed pseudorandom walks, these are walks in which once a starting point is chosen, its path is completely determined. We apply our method to construct a parallel framework for circuit synthesis, and implement one such version performing optimal T -count synthesis over the Clifford+ T gate set. We use our software to present examples where parallelization offers a significant speedup on the runtime, as well as directly confirm that the 4-qubit 1-bit full adder has optimal T -count 7 and T -depth 3.