QW-HHL: A Quantum Computer Amenable General Matrix Equation Solver

We introduce quantum utility , a new approach to evaluating quantum performance that aims to capture the user experience by considering the overhead costs associated with a quantum computation. A demonstration of quantum utility by the quantum processing unit (QPU) shows that the QPU can outperform classical solvers at some tasks of interest to practitioners, when considering the costs of computational overheads. A milestone is a test of quantum utility that is restricted to a specific subset of overhead costs and input types. We illustrate this approach with a benchmark study of a D-Wave annealing-based QPU versus seven classical solvers for a variety of problems in heuristic optimization. We consider overhead costs that arise in standalone use of the D-Wave QPU (as opposed to a hybrid computation). We define three early milestones on the path to broad-scale quantum utility. Milestone 0 is the purely quantum computation with no overhead costs and is demonstrated implicitly by positive results on other milestones. We evaluate the performance of a D-Wave Advantage QPU with respect to milestones 1 and 2: For milestone 1, the QPU outperformed all classical solvers in 99% of our tests. For milestone 2, the QPU outperformed all classical solvers in 19% of our tests, and the scenarios in which the QPU found success correspond to cases where classical solvers most frequently failed. This approach of isolating subsets of overheads for separate analysis reveals distinct mechanisms in quantum versus classical performance, which explain the observed differences in patterns of success and failure. We present evidence-based arguments that these distinctions bode well for annealing quantum processors to support demonstrations of quantum utility on ever-expanding classes of inputs and with more challenging milestones in the very near future.

One of the most prominent application areas for quantum computers is solving hard constraint satisfaction and optimization problems. However, detailed analyses of the complexity of standard quantum algorithms have suggested that outperforming classical methods for these problems would require extremely large and powerful quantum computers. The quantum approximate optimization algorithm (QAOA) is designed for near-term quantum computers, yet previous work has shown strong limitations on the ability of QAOA to outperform classical algorithms for optimization problems. Here we instead apply QAOA to hard constraint satisfaction problems, where both classical and quantum algorithms are expected to require exponential time. We analytically characterize the average success probability of QAOA on a constraint satisfaction problem commonly studied using statistical physics methods: random k-SAT at the threshold for satisfiability, as the number of variables n goes to infinity. We complement these theoretical results with numerical experiments on the performance of QAOA for small n, which match the limiting theoretical bounds closely. We then compare QAOA with leading classical solvers. For random 8-SAT, we find that for more than 14 quantum circuit layers, QAOA achieves more efficient scaling than the highest-performance classical solver we tested, WalkSATlm. Our results suggest that near-term quantum algorithms for solving constraint satisfaction problems may outperform their classical counterparts. Published by the American Physical Society 2024

In the last few years quantum information has played an increasingly central role in the research activities of many scientists within wide ranging areas of physics, mathematics and computer science as it known to be more efficient than its classical counterpart. The impact and advantages of quantum information protocols emerge in numerous situations. In the case of cryptography, quantum dynamics guarantees secure protocols, and in quantum computation, factorization of large numbers, intractable with classical algorithms, can be solved much faster with a quantum computer. It is now widely believed that quantum information will play a leading role in future technologies.Together with the ongoing development of more efficient schemes to solve both new and old tasks in information science, a great deal of interest has been devoted to selecting suitable physical systems where one could realize these ideas. In particular, the quest for large scale integrability has stimulated an increasing interest in the field of solid state physics. Nanoelectronics seems the natural arena to realize physical implementation of quantum hardware. Qubits made out of solid-state devices, such as spins/charges in quantum dots or superconducting nanocircuits, may offer a greater advantage in this respect because fabrication techniques allow for scalability to a large number of coupled qubits. Recent experimental breakthroughs in semiconducting and superconducting nanostructures constitute the first important steps towards the realization of a solid-state quantum computer. In addition the interest in solid state quantum computation has stimulated a large body of research aimed at understanding the properties of entanglement in solid state systems.This Focus issue of New Journal of Physics gathers together contributions from leading experts in the area of solid state quantum information with the aim of providing a panorama of the most exciting scientific questions currently being investigated in the field.Focus on Solid State Quantum Information ContentsQuasiparticle entanglement: redfinition of the vacuum and reduced density matrix approach P Samuelsson, E Sukhorukov and M Buttiker Pseudospin quantum computation in semiconductor nanostructures V W Scarola, K Park and S Das Sarma Superconducting qubit network with controllable nearest-neighbour coupling M Wallquist, J Lantz, V S Shumeiko and G Wendin Weak coupling Josephson junction as a current probe: effect of dissipation on escape dynamics J M Kivioja, T E Nieminen, J Claudon, O Buisson, F Hekking and J P Pekola The single Cooper-pair box as a charge qubit K Bladh, T Duty, D Gunnarsson and P Delsing Quantum state transfer in arrays of flux qubits A O Lyakhov and C Bruder Spin filling of a quantum dot derived from excited-state spectroscopy L H Willems van Beveren, R Hanson, I T Vink, F H L Koppens, L P Kouwenhoven and L M K Vandersypen Clauser–Horne inequality for the full counting statistics F Taddei, R Fazio and E Prada Recent advances in exciton based quantum information processing in quantum dot nanostructures H J Krenner, S Stufler, M Sabathil, E C Clark, P Ester, M Bichler, G Abstreiter and J J Finley Spatially highly resolved study of AFM scanning tip-quantum dot local interaction S Kicin, A Pioda, T Ihn, M Sigrist, A Fuhrer, K Ensslin, M Reinwald and W Wegscheider Transfer of entanglement from electrons to photons by optical selection rules M Titov, B Trauzette, B Michealis and C W J Beenakker Non-Abelian Chern-Simons models with discrete gauge groups on a lattice B Doucot and L B Ioffe Landau–Zener transitions in qubits controlled by electromagnetic fields Martijn Wubs, Keiji Saito, Sigmund Kohler, Yosuke Kayanuma and Peter Hänggi Phase-slip flux qubits J E Mooij and C J P M Harmans Mediated tunable coupling of flux qubits Alec Maassen van den Brink, A J Berkley and M Yalowsky Divergent beams of nonlocally entangled electrons emitted from hybrid normal-superconducting structures Elsa Prada and Fernando Sols Decoherence from ensembles of two-level fluctuators Josef Schriefl, Yuriy Makhlin, Alexander Shnirman and Gerd Schön Semiconductor quantum dots for electron spin qubits W G van der Wiel, M Stopa, T Kodera, T Hatano and S Tarucha Rosario Fazio, NEST-INFM and Scuola Normale Superiore, Pisa, Italy

Explicit Solutions of Linear Matrix Equations

In this paper, we present a new sparse matrix data format that leads to improved memory coalescing and more efficient sparse matrix-vector multiplication for a wide range of problems on high-throughput architectures such as a GPU. The sparse matrix structure is constructed by sorting the rows based on the row length (defined as the number of non-zero elements in a matrix row) followed by a partition into two ranges, short rows and long rows. Based on this partition, the matrix entries are then transformed into ELLPACK or vectorized compressed sparse row format. In addition, the number of threads are adaptively selected by their row length, in order to balance the workload for each graphics processing unit thread. Several computational experiments are presented to support this approach and the results suggest a notable improvement over a wide range of matrix structures.

The maximum parsimony phylogenetic tree reconstruction problem is NP‐hard, presenting a computational bottleneck for classical computing and motivating the exploration of emerging paradigms like quantum computing. To this end, we design three optimization models compatible with both classical and quantum solvers. Our method directly searches the complete solution space of all possible tree topologies and ancestral states, thereby avoiding the potential biases associated with pre‐constructing candidate internal nodes. Among these models, the branch‐based model drastically reduces the number of variables and explicit constraints through a specific variable definition, providing a novel modeling approach effective not only for phylogenetic tree building but also for other tree problems. The correctness of this model is validated with a classical solver, which obtains solutions that are generally better than those from heuristics on the gene dataset. Moreover, our quantum simulations successfully find the exact optimal solutions for small‐scale instances with rapid convergence, highlighting the potential of quantum computing to offer a new avenue for solving these intractable problems in evolutionary biology.

Fundamental transformations in the basic logic of computing are few and far between. Since the invention of digital computers in the early 1940s, the logic underlying computation has remained the same, even as computing hardware evolved from vacuum tubes to silicon transistors. With the advent of quantum computation, a fundamental transformation is near. Quantum computation is based on a different type of logic: rather than being in one of the two states of a classical bit, a quantum bit or qubit can be in a superposition of two states simultaneously. Operations and measurements on these qubits obey the constraints of quantum mechanics. It is now understood that quantum computers have great power in principle to go beyond classical computers, but that not every application is well suited to implementation on quantum computers. For reasons explained in more detail in the Introduction, scientific problems in chemical and materials sciences are uniquely suited to take advantage of quantum computing in the relatively near future. Indeed, quantum computing offers the best hope to solve many of the most important and difficult problems in this field. For example, quantum materials, such as superconductors and complex magnetic materials, show novel kinds of ordered phases that are natural from the point of view of quantum mechanics but difficult to access via computation on classical computers. Quantum sensors based on solid materials are already widely used but could be greatly improved with insight from quantum computations, as could materials for information technologies. Quantum chemical dynamics is another example of a problem that is intrinsically well suited to studies on quantum computers. Applications of quantum chemical dynamics include catalysis, artificial photosynthesis, and other industrially important processes. Quantum computers exist in the laboratory and are beginning to exceed 50 qubits, which is roughly the size beyond which their behavior cannot be predicted or emulated on present-day classical supercomputers. While a quantum computer of 50 qubits is almost certainly not powerful enough to tackle the major scientific challenges in chemical and materials sciences, some of these major challenges start to become accessible with a few hundred qubits if error rates can be kept small. This roundtable was convened to ask how emerging quantum computers can be applied to major scientific problems in chemical and materials sciences, in light of Basic Energy Sciences’s leading role in these fields and the Department of Energy’s leading role in high-performance scientific computation more generally. The main outcome of the roundtable was a consensus that there are scientific problems of great importance on which emerging quantum computers have the potential for disruptive impact, and where comparable progress is unlikely to occur by other means.

How powerful are Quantum Computers? Despite the prevailing belief that Quantum Computers are more powerful than their classical counterparts, this remains a conjecture backed by little formal evidence. Shor's famous factoring algorithm [Shor97] gives an example of a problem that can be solved efficiently on a quantum computer with no known efficient classical algorithm. Factoring, however, is unlikely to be NP-Hard, meaning that few unexpected formal consequences would arise, should such a classical algorithm be discovered. Could it then be the case that any quantum algorithm can be simulated efficiently classically? Likewise, could it be the case that Quantum Computers can quickly solve problems much harder than factoring? If so, where does this power come from, and what classical computational resources do we need to solve the hardest problems for which there exist efficient quantum algorithms? We make progress toward understanding these questions through studying the relationship between classical nondeterminism and quantum computing. In particular, is there a problem that can be solved efficiently on a Quantum Computer that cannot be efficiently solved using nondeterminism? In this thesis we address this problem from the perspective of sampling problems. Namely, we give evidence that approximately sampling the Quantum Fourier Transform of an efficiently computable function, while easy quantumly, is hard for any classical machine in the Polynomial Time Hierarchy. In particular, we prove the existence of a class of distributions that can be sampled efficiently by a Quantum Computer, that likely cannot be approximately sampled in randomized polynomial time with an oracle for the Polynomial Time Hierarchy. Our work complements and generalizes the evidence given in Aaronson and Arkhipov's work [AA2013] where a different distribution with the same computational properties was given. Our result is more general than theirs, but requires a more powerful quantum sampler.

Quantum computers are expected to offer substantial speedups over their classical counterparts and to solve problems that are intractable for classical computers. Beyond such practical significance, the concept of quantum computation opens up new fundamental questions, among them the issue whether or not quantum computations can be certified by entities that are inherently unable to compute the results themselves. Here we present the first experimental verification of quantum computations. We show, in theory and in experiment, how a verifier with minimal quantum resources can test a significantly more powerful quantum computer. The new verification protocol introduced in this work utilizes the framework of blind quantum computing and is independent of the experimental quantum-computation platform used. In our scheme, the verifier is only required to generate single qubits and transmit them to the quantum computer. We experimentally demonstrate this protocol using four photonic qubits and show how the verifier can test the computer's ability to perform measurement-based quantum computations.

Quantum computers are devices, which allow more efficient solutions of problems as compared to their classical counterparts. As the timeline to developing a quantum-error corrected computer is unclear, the quantum computing community has dedicated much attention to developing algorithms for currently available noisy intermediate-scale quantum computers (NISQ). Thus far, within NISQ, optimization problems are one of the most commonly studied and are quite often tackled with the quantum approximate optimization algorithm (QAOA). This algorithm is best known for computing graph partitions with a maximal separation of edges (MaxCut), but can easily calculate other problems related to graphs. Here, I present a novel quantum optimization algorithm, which uses exponentially less qubits as compared to the QAOA while requiring a significantly reduced number of quantum operations to solve the MaxCut problem. Such an improved performance allowed me to partition graphs with 32 nodes on publicly available 5 qubit gate-based quantum computers without any preprocessing such as division of the graph into smaller subgraphs. These results represent a 40% increase in graph size as compared to state-of-art experiments on gate-based quantum computers such as Google Sycamore. The obtained lower bound is 54.9% on the solution for actual hardware benchmarks and 77.6% on ideal simulators of quantum computers. Furthermore, large-scale optimization problems represented by graphs of a 128 nodes are tackled with simulators of quantum computers, again without any predivision into smaller subproblems and a lower solution bound of 67.9% is achieved. The study presented here paves way to using powerful genetic optimizer in synergy with quantum computers

Supervised machine learning is a popular approach to the solution of many real‐life problems. This approach is characterized by the use of labeled datasets to train algorithms for classifying data or predicting outcomes accurately. The question of the extent to which quantum computation can help improve existing classical supervised learning methods is the subject of intense research in the area of quantum machine learning. The debate centers on whether an advantage can be achieved already with current noisy quantum computer prototypes or it is strictly dependent on the full power of a fault‐tolerant quantum computer. The current proposals can be classified into methods that can be suitably implemented on near‐term quantum computers but are essentially empirical, and methods that use quantum algorithms with a provable advantage over their classical counterparts but only when implemented on the still unavailable fault‐tolerant quantum computer. It turns out that, for the latter class, the benefit offered by quantum computation can be shown rigorously using quantum kernels, whereas the approach based on near‐term quantum computers is very unlikely to bring any advantage if implemented in the form of hybrid algorithms that delegate the hard part (optimization) to the far more powerful classical computers.

The usual solution methods of discretised partial differential equations are based exclusively on matrix-vector multiplications as basis operation. On the one hand, this is supported by the use of sparse matrices (cf. §1.3.2.5); on the other hand, one tries to apply fast iterative methods (e.g., the multigrid method [124], [119, §12]) whose basic steps are matrix-vector multiplications. Krylov methods are based on the same concept. However, there are interesting problems which require the solution of a linear or nonlinear matrix equation1 and cannot be solved via the matrix-vector multiplication. Examples are the linear Lyapunov and Sylvester equations as well as the quadratic Riccati equation, which arise, e.g., in optimal control problems for partial differential equations and in model reduction methods. However, there are interesting problems which require the solution of a linear or nonlinear matrix equation and cannot be solved via the matrix-vector multiplication. Examples are the linear Lyapunov and Sylvester equations as well as the quadratic Riccati equation, which arise, e.g., in optimal control problems for partial differential equations and in model reduction methods. The $$\mathcal{H}$$ -matrix arithmetic allows the solution of these matrix equations efficiently. Here, the use of hierarchical matrix operations and matrix-valued functions is only one part of the solution method. Another essential fact is that the solution $$ X \in \mathbb{R}^{I \times I}(n==\!\!\!\!\!/ \!\!\!\!/I)$$ can be replaced by an $$\mathcal{H}$$ -matrix $$X_{\mathcal{H}}$$ . If one considers the equation $$f(X)\;=\;0$$ as a system of n 2 equation for the n 2 components of X, even an optimal solution method has complexity $$\mathcal{O}(n^2)$$ , since this is linear complexity in a number of unknowns (cf. Remark 1.1). Using traditional techniques, the solution of large-scale matrix equations is not feasible. Only an $$\mathcal{H}$$ -matrix $$X_{\mathcal{H}}$$ with $$\mathcal{O}(n\; \mathrm{log}^*n)$$ data instead of n 2 admits a solution with a cost almost linear with respect to n. Section 15.1 introduces Lyapunov and Sylvester equations and discusses their solution. In Section 15.2 we consider quadratic Riccati equation. An interesting approach uses the matrix version of the sign function from §14.1.1. General nonlinear matrix equations may be solved iteratively by Newton’s method or related methods (cf. Section 15.3). As an example, computing the square root of a positive definite matrix is described in §15.3.1. The influence of the truncation error introduced by H-matrix arithmetic is investigated in Section 15.3.

Stable quantum information in topological systems

Sparse general matrix multiplication (SpGEMM) is a fundamental building block in numerous scientific applications. One critical task of SpGEMM is to compute or predict the structure of the output matrix (i.e., the number of nonzero elements per output row) for efficient memory allocation and load balance, which impact the overall performance of SpGEMM. Existing work either precisely calculates the output structure or adopts upper-bound or sampling-based methods to predict the output structure. However, these methods either take much execution time or are not accurate enough. In this paper, we propose a novel sampling-based method with better accuracy and low costs compared to the existing sampling-based method. The proposed method first predicts the compression ratio of SpGEMM by leveraging the number of intermediate products (denoted as FLOP) and the number of nonzero elements (denoted as NNZ) of the same sampled result matrix. And then, the predicted output structure is obtained by dividing the FLOP per output row by the predicted compression ratio. We also propose a reference design of the existing sampling-based method with optimized computing overheads to demonstrate the better accuracy of the proposed method. We construct 623 test cases with various matrix dimensions and sparse structures to evaluate the prediction accuracy. Experimental results show that the absolute relative errors of the proposed method and the reference design are 1.30% and 7.93%, respectively, on average, and 25% and 158%, respectively, in the worst case.

Recent advancements in quantum computing and quantum-inspired algorithms have sparked renewed interest in binary optimization. These hardware and software innovations promise to revolutionize solution times for complex problems. In this work, we propose a novel method for solving linear systems. Our approach leverages binary optimization, making it particularly well-suited for problems with large condition numbers. We transform the linear system into a binary optimization problem, drawing inspiration from the geometry of the original problem and resembling the conjugate gradient method. This approach employs conjugate directions that significantly accelerate the algorithm’s convergence rate. Furthermore, we demonstrate that by leveraging partial knowledge of the problem’s intrinsic geometry, we can decompose the original problem into smaller, independent sub-problems. These sub-problems can be efficiently tackled using either quantum or classical solvers. Although determining the problem’s geometry introduces some additional computational cost, this investment is outweighed by the substantial performance gains compared to existing methods.