Optimization of quantum programs or circuits is a fundamental problem in quantum computing and remains a major challenge. State-of-the-art quantum circuit optimizers rely on heuristics and typically require superlinear, and even exponential, time. Recent work [8] proposed a new approach that pursues a weaker form of optimality called local optimality. Parameterized by a natural number Ω, local optimality insists that each and every Ω-segment of the circuit is optimal with respect to an external optimizer, called the oracle. Local optimization can be performed using only a linear number of calls to the oracle but still incurs quadratic computational overheads in addition to oracle calls. Perhaps most importantly, the algorithm is sequential.
Optimization of quantum programs or circuits is a fundamental problem in quantum computing and remains a major challenge. State-of-the-art quantum circuit optimizers rely on heuristics and typically require superlinear, and even exponential, time. Recent work [8] proposed a new approach that pursues a weaker form of optimality called local optimality. Parameterized by a natural number Ω, local optimality insists that each and every Ω-segment of the circuit is optimal with respect to an external optimizer, called the oracle. Local optimization can be performed using only a linear number of calls to the oracle but still incurs quadratic computational overheads in addition to oracle calls. Perhaps most importantly, the algorithm is sequential.
Recent advances in quantum architectures and computing have motivated the development of new optimizing compilers for quantum programs or circuits. Even though steady progress has been made, existing quantum optimization techniques remain asymptotically and practically inefficient and are unable to offer guarantees on the quality of the optimization. Because many global quantum circuit optimization problems belong to the complexity class QMA (the quantum analog of NP), it is not clear whether quality and efficiency guarantees can both be achieved. In this paper, we present optimization techniques for quantum programs that can offer both efficiency and quality guarantees. Rather than requiring global optimality, our approach relies on a form of local optimality that requires each and every segment of the circuit to be optimal. We show that the local optimality notion can be attained by a cut-and-meld circuit optimization algorithm. The idea behind the algorithm is to cut a circuit into subcircuits, optimize each subcircuit independently by using a specified "oracle" optimizer, and meld the subcircuits by optimizing across the cuts lazily as needed. We specify the algorithm and prove that it ensures local optimality. To prove efficiency, we show that, under some assumptions, the main optimization phase of the algorithm requires a linear number of calls to the oracle optimizer. We implement and evaluate the local-optimality approach to circuit optimization and compare with the state-of-the-art optimizers. The empirical results show that our cut-and-meld algorithm can outperform existing optimizers significantly, by more than an order of magnitude on average, while also slightly improving optimization quality. These results show that local optimality can be a relatively strong optimization criterion and can be attained efficiently.
Recent advances in quantum architectures and computing have motivated the development of new optimizing compilers for quantum programs or circuits. Even though steady progress has been made, existing quantum optimization techniques remain asymptotically and practically inefficient. Because many global quantum circuit optimization problems belong to the complexity class QMA (the quantum analog of NP) [51], it is not clear whether efficiency guarantees can be achieved. In this paper, we present an algorithm that achieves efficiency by aiming at a form of local optimality rather than global optimality. Our local-optimization algorithm cuts a circuit into subcircuits, optimizes each subcircuit with the specified oracle optimizer, and melds the subcircuits by optimizing across the cuts lazily as needed. To prove efficiency, we show that, under some assumptions, the main optimization phase of the algorithm requires a linear number of calls to the oracle optimizer. Our empirical results show that our local-optimization algorithm can outperform existing optimizers significantly, by more than an order of magnitude on average. Perhaps surprisingly, the algorithm achieves these improvements without noticeable degradation in optimization quality. These results show that local optimization can be effective for optimizing quantum circuits.
Optimizing and simulating quantum circuits at scale are critical bottlenecks in realizing the full potential of quantum computing. This thesis delivers a comprehensive suite of systems to overcome these limitations, spanning both physical compilation and classical simulation. For quantum circuit optimization, we first automate the discovery and verification of transformation rules by introducing Equivalence Circuit Class (ECC) sets and an efficient generation algorithm for arbitrary gate sets. We deploy these rules within the superoptimizer Quartz, which utilizes a cost-based backtracking search to escape local minima that trap traditional rule-based methods. To eliminate the exponential runtime penalties inherent to pure search, we introduce QALM, a hybrid optimization framework. By unifying search-based exploration with greedy rule-based exploitation in an interleaved control loop, QALM dynamically escapes local minima without exhaustive backtracking. Evaluated across a large benchmark suite of 248 diverse circuits, QALM outperforms existing search-based optimizers in final circuit quality while operating with the computational efficiency of rule-based systems. While the prior two approaches aim for global optimization, this problem is intrinsically QMA-hard, creating a bottleneck for large programs. To circumvent this issue and scale up, we introduce OAC, a cut-and-meld circuit optimization algorithm. OAC cuts a circuit into subcircuits, applies an existing oracle optimizer independently, and seamlessly melds the results. This algorithm operates with a linear number of oracle calls while attaining local optimality. Empirical evaluation shows that OAC significantly improves the efficiency of state-of-the-art optimizers while enhancing overall quality. Beyond physical execution, the scalable simulation of quantum circuits on classical hardware presents another major challenge. To address this, we present Atlas, a distributed GPU-based simulator that hierarchically partitions circuits to exploit data parallelism while minimizing communication overhead. By leveraging integer linear programming to allocate structurally related gates to nearby GPUs and dynamic programming for kernel scheduling, Atlas runs significantly faster than prior state-of-the-art GPU simulators. Together, these frameworks provide a robust toolchain, improving both the execution of quantum circuits on physical devices and their scalable classical simulation.
Grover’s algorithm is one of the primary algorithms offered as evidence that quantum computers can provide an advantage over classical computers. It involves an “oracle” (external quantum subroutine), which must be specified for a given application and whose internal structure is not part of the formal scaling of the quadratic quantum speedup guaranteed by the algorithm. Grover’s algorithm also requires exponentially many calls to the quantum oracle (approximately 2 n calls where n is the number of qubits) to succeed, raising the question of its implementation on both noisy and error-corrected quantum computers. In this work, we construct a quantum-inspired algorithm executable on a classical computer that performs Grover’s task in a linear number of calls to (simulations of) the oracle—an exponentially smaller number than Grover’s algorithm—and demonstrate this algorithm explicitly for Boolean satisfiability problems. The complexity of our algorithm depends on the cost to simulate the oracle once, which may or may not be exponential, depending on its internal structure. Indeed, Grover’s algorithm does not have an quantum speedup as soon as one is given access to the “source code” of the oracle, which may reveal an internal structure of the problem. Our findings illustrate this point explicitly, as our algorithm exploits the structure of the quantum circuit used to program the quantum computer to speed up the search. There are still problems where Grover’s algorithm would provide an asymptotic speedup if it could be run accurately for large enough sizes. Our quantum-inspired algorithm provides lower bounds, in terms of the quantum-circuit complexity, for the quantum hardware to beat classical approaches for these problems. These estimates, combined with the unfavorable scaling of the success probability of Grover’s algorithm, which in the presence of noise decays as the exponential of the exponential of the number of qubits, makes a practical speedup unrealistic even under extremely optimistic assumptions of the evolution of both hardware quality and availability. Published by the American Physical Society 2024
We consider a generic framework of optimization algorithms based on gradient descent. We develop a quantum algorithm that computes the gradient of a multi-variate real-valued function f : Rd → R by evaluating it at only a logarithmic number of times in superposition. Our algorithm is an improved version of Jordan's gradient computation algorithm [28], providing an approximation of the gradient ▽f with quadratically better dependence on the evaluation accuracy of f, for an important class of smooth functions. Furthermore, we show that objective functions arising from variational quantum circuits usually satisfy the necessary smoothness conditions, hence our algorithm provides a quadratic improvement in the complexity of computing their gradient. We also show that in a continuous phase-query model, our gradient computation algorithm has optimal query complexity up to poly-logarithmic factors, for a particular class of smooth functions. Moreover, we show that for low-degree multivariate polynomials our algorithm can provide exponential speedups compared to Jordan's algorithm in terms of the dimension d.One of the technical challenges in applying our gradient computation procedure for quantum optimization problems is the need to convert between a probability oracle (which is common in quantum optimization procedures) and a phase oracle (which is common in quantum algorithms) of the objective function f. We provide efficient subroutines to perform this delicate interconversion between the two types of oracles incurring only a logarithmic overhead, which might be of independent interest. Finally, using these tools we improve the runtime of prior approaches for training quantum auto-encoders, variational quantum eigensolvers (VQE), and quantum approximate optimization algorithms (QAOA).
Solving combinatorial optimization problems on near-term quantum computers mainly relies on their quadratic unconstrained binary formulation. Lately, binary polynomial formulations start to take the center stage, having a better scaling with respect to the number of used qubits. However, while quadratic forms are straightforward to implement on quantum circuits, polynomial forms of degree d incur in circuit depth overheads, due to the lack of d-qubit gates, for d≥3, on current hardware. In this paper, we study the problem of designing the shallowest quantum circuit for a given polynomial form. We formulate it as a classical binary linear problem and we prove the \textsf{NP}-completeness of a variant of it. On the algorithmic side, we offer a scalable and performant heuristic method to tackle it, Tempura, as well as faster binary linear problem formulations, which offer computational feasibility while sacrificing optimality. Our performance evaluation shows that the algorithms yield far shorter circuits than the state-of-the-art approaches, and Tempura is mostly comparable with them in terms of runtime.