arXiv subjects

Raouf Dridi

Publications and source records attributed to Raouf Dridi.

At least 19 recordsLinked to original sources

arXiv2023

Zeno-effect Computation: Opportunities and Challenges

Adiabatic quantum computing has demonstrated how quantum Zeno can be used to construct quantum optimisers. However, much less work has been done to understand how more general Zeno effects could be used in a similar setting. We use a construction based on three state systems rather than directly in qubits, so that a qubit can remain after projecting out one of the states. We find that our model of computing is able to recover the dynamics of a transverse field Ising model, several generalisations are possible, but our methods allow for constraints to be implemented non-perturbatively and does not need tunable couplers, unlike simple transverse field implementations. We further discuss how to implement the protocol physically using methods building on STIRAP protocols for state transfer. We find a substantial challenge, that settings defined exclusively by measurement or dissipative Zeno effects do not allow for frustration, and in these settings pathological spectral features arise leading to unfavorable runtime scaling. We discuss methods to overcome this challenge for example including gain as well as loss as is often done in an optical setting.

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arXiv2023

Grover Speedup from Many Forms of the Zeno Effect

It has previously been established that adiabatic quantum computation, operating based on a continuous Zeno effect due to dynamical phases between eigenstates, is able to realise an optimal Grover-like quantum speedup. In other words, is able to solve an unstructured search problem with the same $\sqrt{N}$ scaling as Grover's original algorithm. A natural question is whether other manifestations of the Zeno effect can also support an optimal speedup in a physically realistic model (through direct analogue application rather than indirectly by supporting a universal gateset). In this paper we show that they can support such a speedup, whether due to measurement, decoherence, or even decay of the excited state into a computationally useless state. Our results also suggest a wide variety of methods to realise speedup which do not rely on Zeno behaviour. We group these algorithms into three families to facilitate a structured understanding of how speedups can be obtained: one based on phase kicks, containing adiabatic computation and continuous-time quantum walks; one based on dephasing and measurement; and finally one based on destruction of the amplitude within the excited state, for which we are not aware of any previous results. These results suggest that there may be exciting opportunities for new paradigms of analog quantum computing based on these effects.

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arXiv2022

Suppressing unwanted fluctuations in QAOA and approximate quantum annealing

The quantum approximate optimisation algorithm (QAOA) was partially inspired by digitising quantum annealing. Based on this inspiration, we develop techniques to use the additional flexibility of a universal gate-model quantum computer to mitigate fluctuation effects which are known to distort the search space within quantum annealing and lead to false minima. We find that even just the added ability to take Pauli X measurements allows us to modify the mixer angles to counteract these effects by scaling mixer terms in a way proportional to the diagonal elements of the Fubini-Study metric. We find that mitigating these effects can lead to higher success probabilities in cases where the energy landscape is distorted and that we can use the same Pauli X measurements to target which variables are likely to be susceptible to strong fluctuations. The effects of the methods we introduce are relevant even at relatively low depth of $p=10-20$, suggesting that the techniques we are developing are likely to be relevant in the near term. Furthermore, since these methods rely on controlling a degree of freedom which is not typically modified in QAOA, our methods will be compatible with a wide range of other QAOA innovations. We further verify that these fluctuation effects can be observed on an IonQ Harmony QPU.

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arXiv2022

Quantum Natural Gradient with Efficient Backtracking Line Search

We consider the Quantum Natural Gradient Descent (QNGD) scheme which was recently proposed to train variational quantum algorithms. QNGD is Steepest Gradient Descent (SGD) operating on the complex projective space equipped with the Fubini-Study metric. Here we present an adaptive implementation of QNGD based on Armijo's rule, which is an efficient backtracking line search that enjoys a proven convergence. The proposed algorithm is tested using noisy simulators on three different models with various initializations. Our results show that Adaptive QNGD dynamically adapts the step size and consistently outperforms the original QNGD, which requires knowledge of optimal step size to {perform competitively}. In addition, we show that the additional complexity involved in performing the line search in Adaptive QNGD is minimal, ensuring the gains provided by the proposed adaptive strategy dominates any increase in complexity. Additionally, our benchmarking demonstrates that a simple SGD algorithm (implemented in the Euclidean space) equipped with the adaptive scheme above, can yield performances similar to the QNGD scheme with optimal step size. Our results are yet another confirmation of the importance of differential geometry in variational quantum computations. As a matter of fact, we foresee advanced mathematics to play a prominent role in the NISQ era in guiding the design of faster and more efficient algorithms.

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arXiv2022

Understanding domain-wall encoding theoretically and experimentally

We analyze the method of encoding pairwise interactions of higher-than-binary discrete variables (these models are sometimes referred to as discrete quadratic models) into binary variables based on domain walls on one dimensional Ising chains. We discuss how this is relevant to quantum annealing, but also many gate model algorithms such as VQE and QAOA. We theoretically show that for problems of practical interest for quantum computing and assuming only quadratic interactions are available between the binary variables, it is not possible to have a more efficient general encoding in terms of number of binary variables per discrete variable. We furthermore use a D-Wave Advantage 1.1 flux qubit quantum annealing computer to show that the dynamics effectively freeze later for a domain-wall encoding compared to a traditional one-hot encoding. This second result could help explain the dramatic performance improvement of domain wall over one hot which has been seen in a recent experiment on D-Wave hardware. This is an important result because usually problem encoding and the underlying physics are considered separately, our work suggests that considering them together may be a more useful paradigm. We argue that this experimental result is also likely to carry over to a number of other settings, we discuss how this has implications for gate-model and quantum-inspired algorithms.

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arXiv2020

Constrained-optimization Approach Delivers Superior Classical Performance for Graph Partitioning via Quantum-ready Method

Graph partitioning is one of an important set of well-known compute-intense (NP-hard) graph problems that devolve to discrete constrained optimization. We sampled solutions to the problem via two different quantum-ready methods to understand the pros and cons of each method. First we formulated and sampled the problem as a quadratic unconstrained binary optimization (QUBO) problem, via the best known QUBO formulation, using a best-in-class QUBO sampler running purely classically. Second, we formulated the problem at a higher level, as a set of constraints and an objective function, and sampled it with a constrained-optimization sampler (which internally samples the problem via QUBOs also sampled classically). We find that both approaches often deliver better partitions than the purpose-built classical graph partitioners. Further, we find that the constrained-optimization approach is often able to deliver better partitions in less time than the bespoke-QUBO approach, without knowledge of the graph-partitioning problem. Stepping back from graph partitioning itself, one key controversial question is whether bespoke algorithms or general tools are more likely to deliver the power of QCs to real-world users. These results bear on that question, though they require confirmation on other problems and instances as well as replacement of the low-level sampler by a QC. Still, this early evidence supports the proposition that general tools will contribute significantly to a range of problems, expanding the impact of QCs. This benefit is independent of the low-level sampler employed, whether software or QC, so reinforces the need for more work on high-level optimization. An early version of such software is commercially available in the cloud today, delivering superior classical performance for some problems, enables quantum-forward organizations to migrate to quantum-ready methods now.

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arXiv2020

QCI Qbsolv Delivers Strong Classical Performance for Quantum-Ready Formulation

Many organizations that vitally depend on computation for their competitive advantage are keen to exploit the expected performance of quantum computers (QCs) as soon as quantum advantage is achieved. The best approach to deliver hardware quantum advantage for high-value problems is not yet clear. This work advocates establishing quantum-ready applications and underlying tools and formulations, so that software development can proceed now to ensure being ready for quantum advantage. This work can be done independently of which hardware approach delivers quantum advantage first. The quadratic unconstrained binary optimization (QUBO) problem is one such quantum-ready formulation. We developed the next generation of qbsolv, a tool that is widely used for sampling QUBOs on early QCs, focusing on its performance executing purely classically, and deliver it as a cloud service today. We find that it delivers highly competitive results in all of quality (low energy value), speed (time to solution), and diversity (variety of solutions). We believe these results give quantum-forward users a reason to switch to quantum-ready formulations today, reaping immediate benefits in performance and diversity of solution from the quantum-ready formulation,preparing themselves for quantum advantage, and accelerating the development of the quantum computing ecosystem.

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arXiv2019

Integer programming techniques for minor-embedding in quantum annealers

A major limitation of current generations of quantum annealers is the sparse connectivity of manufactured qubits in the hardware graph. This technological limitation generated considerable interest, motivating efforts to design efficient and adroit minor-embedding procedures that bypass sparsity constraints. In this paper, starting from a previous equational formulation by Dridi et al. (arXiv:1810.01440), we propose integer programming (IP) techniques for solving the minor-embedding problem. The first approach involves a direct translation from the previous equational formulation to IP, while the second decomposes the problem into an assignment master problem and fiber condition checking subproblems. The proposed methods are able to detect instance infeasibility and provide bounds on solution quality, capabilities not offered by currently employed heuristic methods. We demonstrate the efficacy of our methods with an extensive computational assessment involving three different families of random graphs of varying sizes and densities. The direct translation as a monolithic IP model can be solved with existing commercial solvers yielding valid minor-embeddings, however, is outperformed overall by the decomposition approach. Our results demonstrate the promise of our methods for the studied benchmarks, highlighting the advantages of using IP technology for minor-embedding problems.

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arXiv2019

The Topology of Mutated Driver Pathways

Much progress has been made, and continues to be made, towards identifying candidate mutated driver pathways in cancer. However, no systematic approach to understanding how candidate pathways relate to each other for a given cancer (such as Acute myeloid leukemia), and how one type of cancer may be similar or different from another with regard to their respective pathways (Acute myeloid leukemia vs. Glioblastoma multiforme for instance), has emerged thus far. Our work attempts to contribute to the understanding of {\em space of pathways} through a novel topological framework. We illustrate our approach, using mutation data (obtained from TCGA) of two types of tumors: Acute myeloid leukemia (AML) and Glioblastoma multiforme (GBM). We find that the space of pathways for AML is homotopy equivalent to a sphere, while that of GBM is equivalent to a genus-2 surface. We hope to trigger new types of questions (i.e., allow for novel kinds of hypotheses) towards a more comprehensive grasp of cancer.

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arXiv2019

GAMA: A Novel Algorithm for Non-Convex Integer Programs

Inspired by the decomposition in the hybrid quantum-classical optimization algorithm we introduced in arXiv:1902.04215, we propose here a new (fully classical) approach to solving certain non-convex integer programs using Graver bases. This method is well suited when (a) the constraint matrix $A$ has a special structure so that its Graver basis can be computed systematically, (b) several feasible solutions can also be constructed easily and (c) the objective function can be viewed as many convex functions quilted together. Classes of problems that satisfy these conditions include Cardinality Boolean Quadratic Problems (CBQP), Quadratic Semi-Assignment Problems (QSAP) and Quadratic Assignment Problems (QAP). Our Graver Augmented Multi-seed Algorithm (GAMA) utilizes augmentation along Graver basis elements (the improvement direction is obtained by comparing objective function values) from these multiple initial feasible solutions. We compare our approach with a best-in-class commercially available solver (Gurobi). Sensitivity analysis indicates that the rate at which GAMA slows down as the problem size increases is much lower than that of Gurobi. We find that for several instances of practical relevance, GAMA not only vastly outperforms in terms of time to find the optimal solution (by two or three orders of magnitude), but also finds optimal solutions within minutes when the commercial solver is not able to do so in 4 or 10 hours (depending on the problem class) in several cases.

math.OC↗

arXiv2019

Knuth-Bendix Completion Algorithm and Shuffle Algebras For Compiling NISQ Circuits

Compiling quantum circuits lends itself to an elegant formulation in the language of rewriting systems on non commutative polynomial algebras $\mathbb Q\langle X\rangle$. The alphabet $X$ is the set of the allowed hardware 2-qubit gates. The set of gates that we wish to implement from $X$ are elements of a free monoid $X^*$ (obtained by concatenating the letters of $X$). In this setting, compiling an idealized gate is equivalent to computing its unique normal form with respect to the rewriting system $\mathcal R\subset \mathbb Q\langle X\rangle$ that encodes the hardware constraints and capabilities. This system $\mathcal R$ is generated using two different mechanisms: 1) using the Knuth-Bendix completion algorithm on the algebra $\mathbb Q\langle X\rangle$, and 2) using the Buchberger algorithm on the shuffle algebra $\mathbb Q[L]$ where $L$ is the set of Lyndon words on $X$.

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arXiv2019

Homological Description of the Quantum Adiabatic Evolution With a View Toward Quantum Computations

We import the tools of Morse theory to study quantum adiabatic evolution, the core mechanism in adiabatic quantum computations (AQC). AQC is computationally equivalent to the (pre-eminent paradigm) of the Gate model but less error-prone, so it is ideally suitable to practically tackle a large number of important applications. AQC remains, however, poorly understood theoretically and its mathematical underpinnings are yet to be satisfactorily identified. Through Morse theory, we bring a novel perspective that we expect will open the door for using such mathematics in the realm of quantum computations, providing a secure foundation for AQC. Here we show that the singular homology of a certain cobordism, which we construct from the given Hamiltonian, defines the adiabatic evolution. Our result is based on E. Witten's construction for Morse homology that was derived in the very different context of supersymmetric quantum mechanics. We investigate how such topological description, in conjunction with Gauß-Bonnet theorem and curvature based reformulation of Morse lemma, can be an obstruction to any computational advantage in AQC. We also explore Conley theory, for the sake of completeness, in advance of any known practical Hamiltonian of interest. We conclude with the instructive case of the ferromagnetic $p-$spin where we show that changing its first order quantum transition (QPT) into a second order QPT, by adding non-stoquastic couplings, amounts to homotopically deform the initial surface accompanied with birth of pairs of critical points. Their number reaches its maximum when the system is fully non-stoquastic. In parallel, the total Gaussian curvature gets redistributed (by the Gauß--Bonnet theorem) around the new neighbouring critical points, which weakens the severity of the QPT.

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arXiv2019

Minimizing polynomial functions on quantum computers

This expository paper reviews some of the recent uses of computational algebraic geometry in classical and quantum optimization. The paper assumes an elementary background in algebraic geometry and adiabatic quantum computing (AQC), and concentrates on presenting concrete examples (with Python codes tested on a quantum computer) of applying algebraic geometry constructs: solving binary optimization, factoring, and compiling. Reversing the direction, we also briefly describe a novel use of quantum computers to compute Groebner bases for toric ideals. We also show how Groebner bases play a role in studying AQC at a fundamental level within a Morse theory framework. We close by placing our work in perspective, by situating this leg of the journey, as part of a marvelous intellectual expedition that began with our ancients over 4000 years ago.

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arXiv2019

Enhancing the efficiency of adiabatic quantum computations

We describe a general methodology for enhancing the efficiency of adiabatic quantum computations (AQC). It consists of homotopically deforming the original "Hamiltonian surface" in a way that the redistribution of the Gaussian curvature weakens the effect of the anti-crossing, thus yielding the desired improvement. Our approach is not pertubative but instead is built on our previous global description of AQC in the language of Morse theory. Through the homotopy deformation we witness the birth and death of critical points whilst, in parallel, the Gauss-Bonnet theorem reshuffles the curvature around the changing set of critical points. Therefore, by creating enough critical points around the anti-crossing, the total curvature--which was initially centered at the original anti-crossing--gets redistributed around the new neighbouring critical points, which weakens its severity and so improves the speedup of the AQC. We illustrate this on two examples taken from the literature.

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arXiv2019

Graver Bases via Quantum Annealing with Application to Non-Linear Integer Programs

We propose a novel hybrid quantum-classical approach to calculate Graver bases, which have the potential to solve a variety of hard linear and non-linear integer programs, as they form a test set (optimality certificate) with very appealing properties. The calculation of Graver bases is exponentially hard (in general) on classical computers, so they not used for solving practical problems on commercial solvers. With a quantum annealer, however, it may be a viable approach to use them. We test two hybrid quantum-classical algorithms (on D-Wave)--one for computing Graver basis and a second for optimizing non-linear integer programs that utilize Graver bases--to understand the strengths and limitations of the practical quantum annealers available today. Our experiments suggest that with a modest increase in coupler precision--along with near-term improvements in the number of qubits and connectivity (density of hardware graph) that are expected--the ability to outperform classical best-in-class algorithms is within reach, with respect to non-linear integer optimization.

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arXiv2018

A Novel Algebraic Geometry Compiling Framework for Adiabatic Quantum Computations

Adiabatic Quantum Computing (AQC) is an attractive paradigm for solving hard integer polynomial optimization problems. Available hardware restricts the Hamiltonians to be of a structure that allows only pairwise interactions. This requires that the original optimization problem to be first converted -- from its polynomial form -- to a quadratic unconstrained binary optimization (QUBO) problem, which we frame as a problem in algebraic geometry. Additionally, the hardware graph where such a QUBO-Hamiltonian needs to be embedded -- assigning variables of the problem to the qubits of the physical optimizer -- is not a complete graph, but rather one with limited connectivity. This "problem graph to hardware graph" embedding can also be framed as a problem of computing a Groebner basis of a certain specially constructed polynomial ideal. We develop a systematic computational approach to prepare a given polynomial optimization problem for AQC in three steps. The first step reduces an input polynomial optimization problem into a QUBO through the computation of the Groebner basis of a toric ideal generated from the monomials of the input objective function. The second step computes feasible embeddings. The third step computes the spectral gap of the adiabatic Hamiltonian associated to a given embedding. These steps are applicable well beyond the integer polynomial optimization problem. Our paper provides the first general purpose computational procedure that can be used directly as a $translator$ to solve polynomial integer optimization. Alternatively, it can be used as a test-bed (with small size problems) to help design efficient heuristic quantum compilers by studying various choices of reductions and embeddings in a systematic and comprehensive manner. An added benefit of our framework is in designing Ising architectures through the study of $\mathcal Y-$minor universal graphs.

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arXiv2016

Prime factorization using quantum annealing and computational algebraic geometry

We investigate prime factorization from two perspectives: quantum annealing and computational algebraic geometry, specifically Gröbner bases. We present a novel scalable algorithm which combines the two approaches and leads to the factorization of all bi-primes up to just over $200 \, 000$, the largest number factored to date using a quantum processor.

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arXiv2016

Homology Computation of Large Point Clouds using Quantum Annealing

Homology is a tool in topological data analysis which measures the shape of the data. In many cases, these measurements translate into new insights which are not available by other means. To compute homology, we rely on mathematical constructions which scale exponentially with the size of the data. Therefore, for large point clouds, the computation is infeasible using classical computers. In this paper, we present a quantum annealing pipeline for computation of homology of large point clouds. The pipeline takes as input a graph approximating the given point cloud. It uses quantum annealing to compute a clique covering of the graph and then uses this cover to construct a Mayer-Vietoris complex. The pipeline terminates by performing a simplified homology computation of the Mayer-Vietoris complex. We have introduced three different clique coverings and their quantum annealing formulation. Our pipeline scales polynomially in the size of the data, once the covering step is solved. To prove correctness of our algorithm, we have also included tests using D-Wave 2X quantum processor.

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