Quantum Approximate Optimisation Applied to Graph Similarity

Nicholas J. Pritchard

REVIEW 4 major objections 5 minor 97 references

Quantum Approximate Optimisation Applied to Graph Similarity

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A compact qubit encoding lets QAOA beat random sampling on small graph-similarity problems, but the zero-cost infeasible tail erodes the advantage on larger instances.

desk verdict A careful QAOA-for-graph-similarity study whose own larger-graph results refute its central claim, made worse by a metric defined with contradictory sign conventions. read the letter →

arxiv 2412.17309 v1 pith:JNPWBJNI submitted 2024-12-23 quant-ph cs.DC

Nicholas J. Pritchard This is my paper · ORCID

classification quant-phcs.DC MSC 05C6068Q1281P68 PACS 03.67.Ac

keywords quantumapproximateoptimisationalgorithmgraphsimilarityedgeoverlappermutationencodingcombinatorialsimulationinfeasiblesolutionsparameter

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish the Quantum Approximate Optimisation Algorithm (QAOA) as a practical tool for whole-graph similarity, measured by maximal edge overlap between two unlabelled directed or undirected graphs. Its route is a compact permutation encoding that uses $\lceil\log_2(V!)\rceil$ qubits instead of $V^2$, plus a custom simulator, Qolab, that tests all components of the algorithm from desktop to cluster scale. The unifying hypothesis is that, for a fixed number of samples, QAOA produces more correct solutions than random sampling. The data support that hypothesis on small graphs and on undirected graphs, while showing the advantage disappears on larger directed instances, and the paper attributes that pattern to the large tail of infeasible zero-cost bit-strings. A working version of this encoding would give near-term quantum computers a memory-efficient route into permutation problems; the measured degradation points instead to where the next encoding or optimiser improvements must go.

What carries the argument

The load-bearing mechanism is the permutation-index encoding: rather than labelling each candidate vertex mapping with $V^2$ unary qubits, the paper orders all $V!$ permutations and stores only the index of a permutation in $q=\lceil\log_2(V!)\rceil$ qubits. The diagonal cost Hamiltonian $\hat C$ gives each index its edge-overlap value, and the unused tail of $2^q-V!$ bit-strings is set to zero. The mixing Hamiltonian $\hat B=\sum_i\sigma^x_i$ makes the search walk over all bit-strings, and $\hat U_B$ is applied through a polynomial expansion of the matrix exponential rather than a full dense exponentiation. Everything that matters for the result follows from that tail: it is what saves memory, and it is what makes the cost landscape hard for the classical optimiser.

What would settle it

Run the exact simulation on randomly deformed graph pairs of increasing vertex count with a fixed evaluation budget, and compare the QAOA expectation value with the expectation of uniform random sampling under the same cost function. The paper's Section 4.4.1 already reports a negative quantum-versus-classical ratio on the larger directed test cases; any negative ratio at a given graph size falsifies the unifying hypothesis for that size, and a consistently positive ratio across the tested range would confirm it.

Watch

Extended reading notes

Core claim

The paper's central claim is that graph similarity by maximal edge overlap can be mapped to QAOA with only $q=\lceil\log_2(V!)\rceil$ qubits, and that on small instances the algorithm measurably improves over naive random sampling for a fixed sample count. The unused bit-strings are assigned zero cost, so the algorithm starts from a uniform superposition in which most amplitude sits on infeasible solutions; nevertheless, the reported experiments show QAOA escaping that local minimum and improving with Trotter depth, with undirected graphs performing better than directed ones. At larger graph sizes the improvement reverses: the paper reports that the QAOA expectation value falls below random sampling for all eight classical optimisers tested, which it treats as evidence that the feasible/infeasible mix makes the optimisation problem harder, not as a final refutation of the approach.

Load-bearing premise

The classical optimiser can concentrate enough probability on feasible permutations within its evaluation budget even though the search starts uniformly over a space in which nearly all bit-strings are infeasible zero-cost strings.

Editorial extensions

If this is right

Small instances and undirected pairs establish a baseline where QAOA with this encoding is a valid laboratory for comparing mixers, cost Hamiltonians, and classical optimisers.
Increased Trotter depth $p$ buys better solutions at roughly double the evaluation count, so any real device must budget quantum evaluations against solution quality.
The zero-cost tail creates a landscape in which all infeasible strings look optimal; any future encoding that shrinks or penalises that tail should directly improve the measured quantum-versus-classical ratio.
Because the simulator is problem-agnostic apart from the cost function, the same package can be used to validate QAOA on other NP-optimisation problems without building a new simulation stack.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The comparison baseline is random sampling only; comparing QAOA on this encoding against classical heuristic graph-matching would clarify whether the small-instance advantage is real or an artefact of a weak baseline.
The paper's alternate cost function that penalises infeasible states made things worse, which suggests a subtle nonlinearity: a soft, tuned penalty or a separate feasible-mixing operator might outperform both tested extremes.
The tail proportion oscillates with $V$ (for example, $2^q/V!$ is 0.65 at $V=8$ and 0.15 at $V=10$ in the paper's Table B.1), so picking graph sizes whose factorial sits close to a power of two could give QAOA an easier landscape and a cleaner test of the algorithm.
The exact state-vector simulation ceiling of 22 qubits is a memory bound, so adapting the encoding to a restricted mixer or approximate simulation scheme would show whether the trend continues or reverses at larger sizes.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a 2018 honours thesis posted to arXiv in 2024. It applies QAOA to graph similarity via edge overlap, uses a compact permutation encoding (credited to Chiew et al.), and presents Qolab, a distributed simulator. The advertised claim is that QAOA beats random sampling for a fixed number of samples. The paper's own results on larger graphs contradict that, and the metric that would decide the question is defined with contradictory sign conventions.

What is genuinely useful: the encoding itself—ceil(log2(V!)) qubits with the tail mapped to zero—is a sensible way to cut qubit count for permutation problems, and the authors are upfront about the infeasible-solution cost. Qolab looks real: state-vector distribution, Lehmer-code permutation generation, Chebyshev-based matrix exponential, and NLopt integration. The optimizer comparison across eight methods and six depths is thorough for a student project. Correctness is checked against brute force, and test cases are generated by graph deformations, which is more careful than many QAOA papers.

Where it falls down: the central hypothesis is not supported. Section 4.4.1 explicitly says that on larger graphs, all optimizers yield a lower expectation value than classical random sampling. That is the opposite of what the abstract promises. Worse, the "Quantum/Classical Expectation" metric in Section 4.1 says "A negative value indicates better likelihood of a better solution from the QAOA. Higher is better." Both clauses cannot be true. The plots show negative values on larger graphs, and the text treats them as failure. This ambiguity decides whether the main result is positive or negative. Additionally, the so-called novel encoding is credited to Chiew et al., so the contribution is the application, not the encoding itself. Resource counts are cited rather than derived, and Appendix E's spectral proof is informal, though the result is correct.

In proportion: the math and data are transparent enough to see what happened, and the writing is clear, but the paper overclaims. A reader cannot tell from the metric definition which direction the evidence points. If the authors reframed the work as an honest negative result about QAOA with tail-heavy encodings, it could be useful. As it stands, the abstract and conclusion are not supported by the data.

Recommendation: I would not publish this as is, but I would send it to peer review if the authors are willing to fix the metric definition and reframe the contribution. The simulator and the empirical optimizer data deserve referee attention, and the negative result, once stated clearly, is worth reporting.

Referee Report

4 major / 5 minor

Summary. The paper presents an honours-thesis-style study applying the Quantum Approximate Optimisation Algorithm (QAOA) to graph similarity, specifically to the problem of maximal edge overlap between two directed or undirected unweighted graphs under vertex relabelling. The authors introduce Qolab, a C-based simulation package for QAOA, and propose a compact encoding of permutation problems that uses q = ceil(log2(V!)) qubits rather than a unary V^2 encoding. The stated unifying hypothesis (Section 1.5) is that, for a fixed number of samples, QAOA will produce more correct maximal-edge-overlap solutions than naive random sampling. The paper reports experiments with eight classical optimisers at six decomposition levels, on graphs generated by Erdős–Rényi models with several deformation types, and concludes that the QAOA suffers from infeasible solutions in larger test cases.

Significance. If the central hypothesis were supported, the paper would offer two useful contributions: a general-purpose QAOA simulation tool with a reproducible benchmark suite, and a memory-efficient encoding for permutation-based optimisation problems. The manuscript does make some genuine assets explicit: the source code and result data are promised at public repositories, the study covers a broad sweep of classical optimisers (eight methods at six values of p), and the correctness of the QAOA expectation is checked against brute-force optimal values in Section 4.1. However, the paper's own reported results contradict the unifying hypothesis for larger graphs, and the primary comparison metric is defined with contradictory sign conventions. The contribution is therefore not established by the evidence presented, and the significance of the claimed result is currently undermined.

major comments (4)

[Section 4.1, 'Classical Comparison' metric] The definition of the 'Quantum / Classical Expectation' metric contains two mutually exclusive clauses: 'A negative value indicates better likelihood of a better solution from the QAOA. Higher is better.' These cannot both hold. If negative means QAOA is better, then higher (less negative) is not better; if higher is better, then negative is worse. The plots in Figures 4.2–4.6 show negative values for larger graphs, and Section 4.4.1 explicitly interprets those negative values as failure ('a lower expectation value versus classical sampling'). This contradiction is load-bearing because the metric is the main evidence for or against the Section 1.5 hypothesis. The paper must state an unambiguous sign convention and then re-interpret the figures consistently.
[Section 4.4.1 and Section 1.5] Section 4.4.1 states: 'for all algorithms tested the effect of infeasible solutions results in a lower expectation value versus classical sampling in the larger test cases.' This is a direct admission that the central hypothesis of Section 1.5 (QAOA outperforms random sampling) is not supported on larger graphs for any of the eight optimisers tested. If the 'negative is better' reading of the metric is instead adopted, then the same sentence would be inconsistent with the reported sign of the values. Either way, the paper does not present an unambiguous positive result, and the hypothesis as stated is not established. This is not a presentation issue but a fundamental problem with the claimed conclusion.
[Section 3.2.4, tail mapping and Section 4.4.1] The encoding maps all 2^q − V! infeasible bit-strings to cost zero, and the QAOA starts from a uniform superposition over all 2^q states. As the paper itself notes and as Table B.1 shows, the tail fraction can be substantial (e.g., for V=8, 2^q/V! ≈ 1.63). The paper attributes the degraded expectation values on larger graphs to the 'effect of infeasible solutions,' which is a plausible mechanism, but no evidence is provided that the classical optimiser can overcome this tail within the evaluation budget. Since the Section 1.5 hypothesis depends precisely on the QAOA concentrating probability on feasible permutations, the paper needs either a dedicated analysis of the tail's influence or a revised hypothesis that accounts for the reported negative results.
[Sections 3.2.2 and 3.2.4, resource claims] The claims of 'O(V^3) operations' for the state-preparation encoding and 'O(n^2 log n) operations and O(n log n) qubits' for the edge-overlap circuit are asserted by citation to [1] and [18] rather than derived in this manuscript. Because the claimed 'significant quantum memory savings at the cost of additional operations' is one of the paper's two headline contributions, the resource trade-off should be either derived or stated with the precise conditions under which the cited bounds apply. As written, the reader cannot verify the resource count without going to the cited works, and the paper's own Appendix E, which is used to justify the Chebyshev truncation for U_B, is informal.

minor comments (5)

[Figures 4.2–4.6 and Appendix F] The figures do not include error bars, confidence intervals, or any measure of trial-to-trial variance; given that results are aggregated over different deformation types and multiple random instances, the absence of dispersion information makes it difficult to assess the reliability of the reported averages.
[Section 4.5 and figures] The text in Section 4.5 refers to 'Figure 4.5' for the undirected Subplex results, but the displayed figure caption is 'Figure 4.6: Final performance for the Subplex algorithm (undirected graphs)'. Similar numbering inconsistencies appear across the results chapter and should be corrected.
[Section 3.4.1] The sentence 'This requires the generation of all q! bit-strings' is confusing: the simulation enumerates all 2^q computational basis states, not q! bit-strings. The subsequent sentence about reducing 'generations of the q! permutations' to the 2^{ceil(log2(v!))} state-space needs to be clarified.
[Equation (3.7.3)] The Chebyshev expansion is written with 'inf' instead of an infinity symbol, and the summation index n is not defined in the displayed formula; the truncation criterion in Eq. (3.7.8) should also state explicitly that n runs from 1 to the truncation order.
[Appendix E] The proof of λ_{min,max} = ±q is informal: it uses a real-symmetric-matrix argument with a typo ('R3' instead of R^n), invokes Perron–Frobenius without stating the non-negativity conditions precisely, and the final observation 'for a negative v, λ = −q' does not constitute a proof for all vectors. The paper also contains typos such as 'Schr¨ oidgner' in Section 1.4.1 and 'na ¨ ıve' elsewhere.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central QAOA-vs-random comparison is benchmarked against independent brute-force optimal values and per-instance optimisation, not fitted to its own outputs.

full rationale

The paper's derivation chain is self-contained rather than circular. Section 3.2 constructs the problem Hamiltonian C from the graph edge-overlap objective directly, and Section 3.7-3.8 implements the QAOA state evolution and measurement without any parameter fitted to a target answer. The Section 1.5 hypothesis is tested in Section 4 by comparing the QAOA expectation value with an independent random-sampling baseline and with known brute-force optimal values computed by Algorithm 1; the QAOA angles are optimised per instance by nlopt rather than tuned to reproduce the benchmark results. The only in-group citation, Marsh and Wang [61], supports an optional walk-mask extension and is not load-bearing for the central compact permutation encoding, which is derived from Abrams-Lloyd and Chiew et al. rather than imported from the authors. The problematic sign convention in the Section 4.1 'Classical Comparison' definition ('A negative value indicates better likelihood of a better solution from the QAOA. Higher is better') and the interpretation in Section 4.4.1 that negative values indicate failure are internal-consistency and correctness defects, not circularity, because neither clause makes the reported comparison equal to its inputs by construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central empirical claim depends on the representativeness of Erdos-Renyi deformed graphs, the implementability of the diagonal unitary at the claimed cost, and the ability of classical optimizers to overcome the zero-cost tail. No new physical entities are introduced. The only hand-set numeric parameters are the MLSL stopping multiplier (200), the Chebyshev tolerance (1e-18), and the random-graph edge probability (0.5).

free parameters (3)

MLSL evaluation cap multiplier S = 200
Section 4.4.1 sets MAX = S * P * GraphSize with S nominally 200, described as the standard value used by Scipy. This hand-chosen stopping rule shapes all MLSL correctness and efficiency results.
Chebyshev truncation tolerance epsilon = 1e-18
Section 3.7.3 terminates the Chebyshev expansion when |2Jn(alpha)| is below epsilon, chosen as 1e-18. This numerical tolerance affects runtime and accuracy but is chosen by hand.
Erdos-Renyi edge probability = 0.5
Section 3.2.5 generates each edge with 50% probability. This choice defines the test distribution and no sensitivity analysis is reported.

assumptions (5)

standard math QAOA with cost Hamiltonian C, mixing Hamiltonian B = sum sigma_x, and p Trotter steps approximately optimizes the expectation value F_p(gamma, beta).
Adopted from Farhi et al. [31]; the paper does not derive the algorithm but uses it as the framework in Section 3.1.
domain assumption The edge-overlap cost for all V! permutations can be encoded as a diagonal Hamiltonian, Eq. 3.2.2, and implemented as e^{-i gamma C} without ancillas via Welch et al. and Chiew et al.
The paper cites these constructions for the claimed O(V^3) preparation and no-ancilla implementation, but gives no gate-level circuit or derivation, so the resource claim is assumed.
ad hoc to paper Mapping all 2^q - V! infeasible bit-strings to cost zero still permits the QAOA to concentrate probability on feasible permutations.
Section 3.2.4 introduces the tail mapping; Section 4.4.1 reports that on larger instances the optimizer fails to beat random sampling, so this assumption is not supported by the paper's data.
standard math The mixing Hamiltonian B has eigenvalues exactly +/- q, so the Chebyshev propagator can be scaled without solving an eigenproblem.
This is a known property of the hypercube adjacency matrix, but the proof in Appendix E is informal and the statement is invoked in Section 3.7.3.
domain assumption Erdos-Renyi graphs with edge probability 0.5 and the five graph deformations are representative difficult instances for graph similarity.
Section 3.2.5 defines the test distribution; no comparison with established graph-similarity benchmarks is given, so representativeness is assumed.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Quantum Approximate Optimisation Applied to Graph Similarity." pith.science (2026). https://pith.science/paper/JNPWBJNI

@misc{pith2026241217309,
  author       = {Pith},
  title        = {Pith review of: Quantum Approximate Optimisation Applied to Graph Similarity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JNPWBJNI}},
  note         = {Machine review of arXiv:2412.17309}
}

read the original abstract

Quantum computing promises solutions to classically difficult and new-found problems through controlling the subtleties of quantum computing. The Quantum Approximate Optimisation Algorithm (QAOA) is a recently proposed quantum algorithm designed to tackle difficult combinatorial optimisation problems utilising both quantum and classical computation. The hybrid nature, generality and typically low gate-depth make it a strong candidate for near-term implementation in quantum computing. Finding the practical limits of the algorithm is currently an open problem. Until now, no tools to facilitate the design and validation of probabilistic quantum optimisation algorithms such as the QAOA on a non-trivial scale exist. Graph similarity is a long standing classically difficult problem withstanding decades of research from academia and industry. Determining the maximal edge overlap between all possible node label permutations is an NP-Complete task and provides an apt measure of graph similarity. We introduce a novel quantum optimisation simulation package facilitating investigation of all constituent components of the QAOA from desktop to cluster scale using graph similarity as an example. Our simulation provides flexibility and performance. We investigate eight classical optimisation methods each at six levels of decomposition. Moreover an encoding for permutation based problems such as graph similarity through edge overlap to the QAOA allows for significant quantum memory savings at the cost of additional operations. This compromise extends into the classical portion of the algorithm as the inclusion of infeasible solutions creates a challenging cost-function landscape. We present performance analysis of our simulation and of the QAOA setting a precedent for investigating and validating numerous other difficult problems to the QAOA as we move towards realising practical quantum computation.

Figures

Figures reproduced from arXiv: 2412.17309 by the authors.

**Figure 1.1.** Two example unlabelled graphs (G1 and G2) • The set of problem instances I is the set of all un-weighted graphs. • Given a pair of graphs G1, G2 the set of feasible solutions is the set of mappings between the vertices of G2 to G1. The number of candidate mappings grows O(V 2 ) hence growing non-polynomially. A solution can be validated by checking that each vertex is mapped and that the mapping is bijective. • The… view at source ↗

**Figure 3.1.** Cˆ This provides us with the edge overlap for all V ! permuted labels of u, v ∈ G2. Our problem operator Cˆ |ψ⟩ = C(x)|ψ⟩ that is the application of our cost function to all bit-strings x. Our diagonal problem operator Uˆ C becomes Uˆ C(γ)|ψ⟩ = e −iγc(x) |ψ⟩. (3.2.3) Welch et al. [87] builds on the work of Childs [19] providing a method to implement e −iγCˆ efficiently without the use of additional ancillary qubits… view at source ↗

**Figure 3.2.** Bˆ 3.2.4 Permutation Mapping Abrams and Lloyd [1] provide a method to encode a superposition of n elements using O(n 3 ) operations. Chiew et al. [18] provide a full circuit encoding edge overlap values using O(n 2 log2 (n)) operations and O(nlog(n)) qubits allowing efficient bit-string mapping over the range [0, n!] to edge-overlap values. To index all such values in Uˆ C we require ⌈log2(V !)⌉ qubits (termed q) s… view at source ↗

Figures from the paper (12 more)

**Figure 3.3.** Figure 3.3: The proportion of infeasible to feasible bit-strings [PITH_FULL_IMAGE:figures/full_fig_p038_3_3.png]

**Figure 3.4.** Figure 3.4: Adjacency Flip state-space. Matrix operations are handled using Intel’s Math Kernel Library [48] and we store matrices in compressed column form. [27]. 3.3.1 Simulation Components Direct state-vector simluation avoids a large amount of overhead present in contempora…

**Figure 4.1.** Figure 4.1: Expectation error for differing cost functions [PITH_FULL_IMAGE:figures/full_fig_p049_4_1.png]

**Figure 4.** Figure 4: depicts performance metrics for the Nelder-Mead algorithm. We see ex [PITH_FULL_IMAGE:figures/full_fig_p049_4.png]

**Figure 4.2.** Figure 4.2: Final performance for the Nelder-Mead algorithm (directed graphs) [PITH_FULL_IMAGE:figures/full_fig_p050_4_2.png]

**Figure 4.3.** Figure 4.3: Final performance for the BOBYQA algorithm (directed graphs) [PITH_FULL_IMAGE:figures/full_fig_p051_4_3.png]

**Figure 4.4.** Figure 4.4: Final performance for the MLSL algorithm (directed graphs) [PITH_FULL_IMAGE:figures/full_fig_p053_4_4.png]

**Figure 4.5.** Figure 4.5: Final performance for the DIRECT algorithm (directed graphs) [PITH_FULL_IMAGE:figures/full_fig_p054_4_5.png]

**Figure 4.6.** Figure 4.6: Final performance for the Subplex algorithm (undirected graphs) [PITH_FULL_IMAGE:figures/full_fig_p055_4_6.png]

**Figure 4.7.** Figure 4.7: Threading performance 4.6 Simulation Performance We plot timing data for four major components of the simulation in [PITH_FULL_IMAGE:figures/full_fig_p056_4_7.png]

**Figure 4.8.** Figure 4.8: Relative speed-up for critical simulation tasks [PITH_FULL_IMAGE:figures/full_fig_p057_4_8.png]

**Figure 4.9.** Figure 4.9: Timing for simulation tasks with optimal threading [PITH_FULL_IMAGE:figures/full_fig_p057_4_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

97 extracted references · 64 canonical work pages

[1]

Simulation of Many-Body Fermi Sys- tems on a Universal Quantum Computer

Daniel S. Abrams and Seth Lloyd. “Simulation of Many-Body Fermi Sys- tems on a Universal Quantum Computer”. In: Physical Review Letters 79.13 (Sept. 29, 1997), pp. 2586–2589. issn: 0031-9007, 1079-7114. doi: 10.1103/ PhysRevLett . 79 . 2586. url: https : / / link . aps . org / doi / 10 . 1103 / PhysRevLett.79.2586 (visited on 10/15/2018)

1997
[18]

Graph comparison via nonlinear quantum search

M. Chiew et al. “Graph comparison via nonlinear quantum search”. In: arXiv:1810.01647 [quant-ph] (Oct. 3, 2018). arXiv: 1810.01647. url: http: //arxiv.org/abs/1810.01647 (visited on 10/15/2018)

arXiv 2018
[2]

Quantum attacks on Bitcoin, and how to protect against them

D. Aggarwal et al. “Quantum attacks on Bitcoin, and how to protect against them”. In: ArXiv e-prints (Oct. 2017)

2017
[3]

A New Scaling and Squar- ing Algorithm for the Matrix Exponential

Awad H. Al-Mohy and Nicholas J. Higham. “A New Scaling and Squar- ing Algorithm for the Matrix Exponential”. In: SIAM Journal on Matrix Analysis and Applications 31.3 (Jan. 2010), pp. 970–989. issn: 0895-4798, 1095-7162. doi: 10.1137/09074721X. url: http://epubs.siam.org/doi/ 10.1137/09074721X (visited on 09/16/2018)

doi:10.1137/09074721x 2010
[4]

Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators

Awad H. Al-Mohy and Nicholas J. Higham. “Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators”. In: SIAM Journal on Scientific Computing 33.2 (Jan. 2011), pp. 488–511. issn: 1064-8275, 1095-7197. doi: 10.1137/100788860. url: http://epubs.siam. org/doi/10.1137/100788860 (visited on 08/02/2018)

doi:10.1137/100788860 2011
[5]

Matrix exponentials and parallel prefix computation in a quantum control problem

T. Auckenthaler et al. “Matrix exponentials and parallel prefix computation in a quantum control problem”. In: Parallel Computing 36.5 (June 2010), pp. 359–369. issn: 01678191. doi: 10.1016/j.parco.2010.01.006 . url: http://linkinghub.elsevier.com/retrieve/pii/S0167819110000244 (visited on 09/17/2018)

work page doi:10.1016/j.parco.2010.01.006 2010
[6]

PETSc Web page

Satish Balay et al. PETSc Web page . 2018. url: http://www.mcs.anl. gov/petsc

2018
[7]

Beating the random assignment on constraint satisfaction problems of bounded degree

Boaz Barak et al. “Beating the random assignment on constraint satisfaction problems of bounded degree”. In: arXiv:1505.03424 [cs] (May 13, 2015). arXiv: 1505.03424. url: http://arxiv.org/abs/1505.03424 (visited on 10/05/2018)

arXiv 2015

Show all 97 references

[8]

Strengths and Weaknesses of Quantum Com- puting

Charles H. Bennett et al. “Strengths and Weaknesses of Quantum Com- puting”. In: SIAM Journal on Computing 26.5 (1997), pp. 1510,1523. issn: 0097-5397. 54

1997
[9]

Efficient computation of the expo- nential operator for large, sparse, symmetric matrices

Luca Bergamaschi and Marco Vianello. “Efficient computation of the expo- nential operator for large, sparse, symmetric matrices”. In: Numerical Linear Algebra with Applications 7.1 (Jan. 2000), pp. 27–45. issn: 1070-5325, 1099-

2000
[10]

Quantum Supremacy Lower Bounds by Entanglement Scaling

Jacob D. Biamonte, Mauro E. S. Morales, and Dax Enshan Koh. “Quantum Supremacy Lower Bounds by Entanglement Scaling”. In: arXiv:1808.00460 [cond-mat, physics:quant-ph] (Aug. 1, 2018). arXiv: 1808.00460. url: http: //arxiv.org/abs/1808.00460 (visited on 08/03/2018)

2018 arXiv
[11]

Characterizing Quantum Supremacy in Near-Term Devices

S. Boixo et al. “Characterizing Quantum Supremacy in Near-Term Devices”. In: ArXiv e-prints (July 2016)

2016
[12]

Simulation of low-depth quantum circuits as complex undi- rected graphical models

S. Boixo et al. “Simulation of low-depth quantum circuits as complex undi- rected graphical models”. In: ArXiv e-prints (Dec. 2017)

2017
[13]

Pseudo-Boolean optimization

Endre Boros and Peter L Hammer. “Pseudo-Boolean optimization”. In: Dis- crete Applied Mathematics (2002), p. 71

2002
[14]

The anatomy of a large-scale hypertextual Web search engine

Sergey Brin and Lawrence Page. “The anatomy of a large-scale hypertextual Web search engine”. In: Computer Networks and ISDN Systems 30.1 (1998), pp. 107,117. issn: 0169-7552

1998
[15]

Published: Cryptology ePrint Archive, Report 2017/847

Andr´ e Chailloux, Mar ´ ıa Naya-Plasencia, and Andr´ e Schrottenloher.An Ef- ficient Quantum Collision Search Algorithm and Implications on Symmetric Cryptography. Published: Cryptology ePrint Archive, Report 2017/847. 2017. url: https://eprint.iacr.org/2017/847

2017
[16]

Classical Simulation of Intermediate-Size Quantum Circuits

J. Chen et al. “Classical Simulation of Intermediate-Size Quantum Circuits”. In: ArXiv e-prints (May 2018)

2018
[17]

64-Qubit Quantum Circuit Simulation

Z.-Y. Chen et al. “64-Qubit Quantum Circuit Simulation”. In: ArXiv e-prints (Feb. 2018)

2018
[19]

Quantum Information Processing in Continu- ous Time

Andrew MacGregor Childs. “Quantum Information Processing in Continu- ous Time”. In: (2004), p. 140

2004
[20]

Quantum machine learning: a classical perspective

Carlo Ciliberto et al. “Quantum machine learning: a classical perspective”. In: Proceedings. Mathematical, Physical, and Engineering Sciences 474.2209 (2018), Proceedings. Mathematical, Physical, and Engineering Sciences, 2018, Vol.474(2209). issn: 1364-5021. 55

2018
[21]

Quantum algorithms revisited

R. Cleve et al. “Quantum algorithms revisited”. In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences454.1969 (1998), pp. 339,354. issn: 1364-5021

1998
[22]

Different Strategies for Optimization Using the Quantum Adiabatic Algorithm

E. Crosson et al. “Different Strategies for Optimization Using the Quantum Adiabatic Algorithm”. In: ArXiv e-prints (Jan. 2014)

2014
[23]

Quantum Theory, the Church-Turing Principle and the Uni- versal Quantum Computer

D. Deutsch. “Quantum Theory, the Church-Turing Principle and the Uni- versal Quantum Computer”. In: Proceedings of the Royal Society of Lon- don. Series A, Mathematical and Physical Sciences (1934-1990) 400.1818 (), pp. 97,117. issn: 0080-4630

1934
[24]

Rapid Solution of Problems by Quan- tum Computation

David Deutsch and Richard Jozsa. “Rapid Solution of Problems by Quan- tum Computation”. In: Proceedings of the Royal Society: Mathematical and Physical Sciences (1990-1995) 439.1907 (1992), pp. 553,558. issn: 0962-8444

1992
[25]

Reinhard. Diestel. Graph theory. 2nd ed. Graduate texts in mathematics
[26]

A new notation for quantum mechanics

P. A. M. Dirac. “A new notation for quantum mechanics”. In: Mathematical Proceedings of the Cambridge Philosophical Society 35.3 (July 1939), p. 416. issn: 0305-0041, 1469-8064. doi: 10.1017/S0305004100021162. url: http: //www.journals.cambridge.org/abstract_S0305004100021162 ...

1939 doi
[27]

Sparse matrix test problems

I Duff, Roger Grimes, and John Lewis. “Sparse matrix test problems”. In: ACM Transactions on Mathematical Software (TOMS) 15.1 (Mar. 1, 1989), pp. 1,14. issn: 1557-7295

1989
[28]

One step time propagation method for systems with time- dependent Hamiltonians

Jian-Yun Fang. “One step time propagation method for systems with time- dependent Hamiltonians”. In: Chemical Physics Letters 263.6 (Dec. 1996), pp. 759–766. issn: 00092614. doi: 10.1016/S0009-2614(96)01272-9 . url: http://linkinghub.elsevier.com/retrieve/pii/S0009261496012729...

1996 doi
[29]

A Quantum Approximate Opti- mization Algorithm Applied to a Bounded Occurrence Constraint Problem

E. Farhi, J. Goldstone, and S. Gutmann. “A Quantum Approximate Opti- mization Algorithm Applied to a Bounded Occurrence Constraint Problem”. In: ArXiv e-prints (Dec. 2014)

2014
[30]

Quantum Supremacy through the Quantum Approximate Optimization Algorithm

E. Farhi and A. W Harrow. “Quantum Supremacy through the Quantum Approximate Optimization Algorithm”. In: ArXiv e-prints (Feb. 2016)

2016
[31]

A Quantum Approx- imate Optimization Algorithm

Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. “A Quantum Approx- imate Optimization Algorithm”. In: arXiv:1411.4028 [quant-ph] (Nov. 14, 2014). arXiv: 1411.4028. url: http://arxiv.org/abs/1411.4028 (visited on 10/05/2018). 56

2014 arXiv
[32]

A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem

Edward Farhi et al. “A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem”. In: Science 292.5516 (2001), pp. 472,5. issn: 00368075. url: http://search.proquest.com/docview/ 213575154/?pq-origsite=primo

2001
[33]

Simulating physics with computers

Richard Feynman. “Simulating physics with computers”. In: International Journal of Theoretical Physics 21.6 (1982), pp. 467,488. issn: 0020-7748

1982
[34]

Michael R. Garey. Computers and intractability : a guide to the theory of NP- completeness. A Series of books in the mathematical sciences. San Francisco: W. H. Freeman, 1979. isbn: 0-7167-1044-7

1979
[35]

A fast quantum mechanical algorithm for database search

L. K. Grover. “A fast quantum mechanical algorithm for database search”. In: eprint arXiv:quant-ph/9605043 (May 1996)

1996 arXiv
[36]

How fast can a quantum computer search?

L. K. Grover. “How fast can a quantum computer search?” In: eprint arXiv:quant- ph/9809029 (Sept. 1998)

1998
[37]

Practical optimization for hybrid quantum-classical algorithms

Gian Giacomo Guerreschi and Mikhail Smelyanskiy. “Practical optimization for hybrid quantum-classical algorithms”. In: arXiv:1701.01450 [quant-ph] (Jan. 5, 2017). arXiv: 1701.01450 . url: http://arxiv.org/abs/1701. 01450 (visited on 10/05/2018)

2017 arXiv
[38]

Quantum Algorithms for Scientific Computing and Approxi- mate Optimization

S. Hadfield. “Quantum Algorithms for Scientific Computing and Approxi- mate Optimization”. In: ArXiv e-prints (May 2018)

2018
[39]

From the Quantum Approximate Optimization Algo- rithm to a Quantum Alternating Operator Ansatz

S. Hadfield et al. “From the Quantum Approximate Optimization Algo- rithm to a Quantum Alternating Operator Ansatz”. In:ArXiv e-prints (Sept. 2017)

2017
[40]

Topological Similarity-Based Feature Selection for Graph Classification

Yongkoo Han et al. “Topological Similarity-Based Feature Selection for Graph Classification.” In: The Computer Journal 58.9 (2015), pp. 1884–1893. doi: 10.1093/comjnl/bxt123 . url: http://dx.doi.org/10.1093/comjnl/ bxt123

2015 doi
[41]

Development of a chemical structure comparison method for integrated analysis of chemical and genomic information in the metabolic pathways

M Hattori et al. “Development of a chemical structure comparison method for integrated analysis of chemical and genomic information in the metabolic pathways”. In: Journal Of The American Chemical Society 125.39 (2003), pp. 11853,11865. issn: 0002-7863

2003
[42]

Solving the graph-isomorphism problem with a quantum annealer

Itay Hen and A. P. Young. “Solving the graph-isomorphism problem with a quantum annealer”. In: Physical Review A 86.4 (Oct. 10, 2012). issn: 1050- 2947, 1094-1622. doi: 10.1103/PhysRevA.86.042310. url: https://link. aps.org/doi/10.1103/PhysRevA.86.042310 (visited on 10/15/2018)

2012 doi
[43]

SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems

Vicente Hernandez, Jose E. Roman, and Vicente Vidal. “SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems”. In: ACM Trans. Math. Software 31.3 (2005), pp. 351–362. 57

2005
[44]

Deriving phylogenetic trees from the similarity analysis of metabolic pathways

Maureen Heymans and Ambuj K. Singh. “Deriving phylogenetic trees from the similarity analysis of metabolic pathways”. In:Bioinformatics 19.1 (2003), pp. 138,146. issn: 1367-4803

2003
[45]

A Block Algorithm for Matrix 1- Norm Estimation, with an Application to 1-Norm Pseudospectra

Nicholas J. Higham and Fran¸ coise Tisseur. “A Block Algorithm for Matrix 1- Norm Estimation, with an Application to 1-Norm Pseudospectra”. In: SIAM Journal on Matrix Analysis and Applications 21.4 (Jan. 2000), pp. 1185–

2000
[46]

0.5 Petabyte Simulation of a 45- Qubit Quantum Circuit

Thomas H¨ aner and Damian S. Steiger. “0.5 Petabyte Simulation of a 45- Qubit Quantum Circuit”. In: arxiv1704.01127 (2017), Proceedings of the In- ternational Conference for High Performance Computing, Networking, Stor- age and Analysis. SC 2017. Article No. 33

2017 arXiv
[47]

Qiskit Aqua

IBM. Qiskit Aqua. url: https://github.com/Qiskit/aqua
[48]

url: https : / / software

Intel(R) Math Kernel Library 2018 Update 4 . url: https : / / software . intel.com/en-us/mkl

2018
[49]

pyCTQW: A continuous-time quantum walk simulator on distributed memory computers

Josh A. Izaac and Jingbo B. Wang. “pyCTQW: A continuous-time quantum walk simulator on distributed memory computers”. In: Computer Physics Communications 186 (Jan. 2015), pp. 81–92. issn: 00104655. doi: 10.1016/ j.cpc.2014.09.011. url: http://linkinghub.elsevier.com/retrieve/ ...

2015
[50]

Six Degrees of Information Seeking: Stanley Milgram and the Small World of the Library

Kathryn James. “Six Degrees of Information Seeking: Stanley Milgram and the Small World of the Library”. In: Journal of Academic Librarianship 32.5 (2006), pp. 527,532. issn: 0099-1333

2006
[51]

Steven G. Johnson. The NLopt nonlinear-optimization package . 2011. url: http://ab-initio.mit.edu/nlopt

2011
[52]

Lipschitzian optimization with- out the Lipschitz constant

D. Jones, C. Perttunen, and B. Stuckman. “Lipschitzian optimization with- out the Lipschitz constant”. In: Journal of Optimization Theory and Appli- cations 79.1 (1993), pp. 157,181. issn: 0022-3239

1993
[53]

SciPy: Open source scientific tools for Python

Eric Jones et al. SciPy: Open source scientific tools for Python . 2001. url: http://www.scipy.org/

2001
[54]

A Preview of Bristlecone, Google’s New Quantum Processor

Julian Kelly. A Preview of Bristlecone, Google’s New Quantum Processor
[55]

Authoritative sources in a hyperlinked environment

Jon Kleinberg. “Authoritative sources in a hyperlinked environment”. In: Journal of the ACM 46.5 (1999), pp. 604,632. issn: 0004-5411. url: http: //search.proquest.com/docview/1808072946/. 58

1999
[56]

Fast parallel algorithms for graph similarity and matching

Giorgos Kollias et al. “Fast parallel algorithms for graph similarity and matching”. In: Journal of Parallel and Distributed Computing 74.5 (2014), pp. 2400 –2410. issn: 0743-7315. doi: https://doi.org/10.1016/j.jpdc. 2013.12.010 . url: http://www.sciencedirect.com/science/art...

2014 doi
[57]

Distortion invariant object recognition in the dynamic link architecture

M. Lades et al. “Distortion invariant object recognition in the dynamic link architecture”. In: IEEE Transactions on Computers 42.3 (Mar. 1993), pp. 300–311. issn: 0018-9340. doi: 10.1109/12.210173

1993 doi
[58]

Quantum Supremacy Circuit Simulation on Sunway Taihu- Light

R. Li et al. “Quantum Supremacy Circuit Simulation on Sunway Taihu- Light”. In: ArXiv e-prints (Apr. 2018)

2018
[59]

Ising formulations of many NP problems

Andrew Lucas. “Ising formulations of many NP problems”. In: Frontiers in Physics 2 (2014). issn: 2296-424X. doi: 10 . 3389 / fphy . 2014 . 00005. arXiv: 1302.5843 . url: http://arxiv.org/abs/1302.5843 (visited on 09/20/2018)

2014 arXiv
[60]

Quantum Supremacy Is Both Closer and Farther than It Appears

Igor L. Markov et al. “Quantum Supremacy Is Both Closer and Farther than It Appears”. In: arXiv:1807.10749 [quant-ph] (July 27, 2018). arXiv: 1807. 10749. url: http://arxiv.org/abs/1807.10749 (visited on 08/01/2018)

2018 arXiv
[61]

A quantum walk assisted approximate algorithm for bounded NP optimisation problems

Samuel Marsh and Jingbo Wang. “A quantum walk assisted approximate algorithm for bounded NP optimisation problems”. In: arXiv:1804.08227 [quant-ph] (Apr. 22, 2018). arXiv: 1804.08227. url: http://arxiv.org/ abs/1804.08227 (visited on 10/15/2018)

2018 arXiv
[62]

Nineteen Dubious Ways to Compute the Exponential of a Matrix

Cleve Moler and Charles Van Loan. “Nineteen Dubious Ways to Compute the Exponential of a Matrix”. In: SIAM Review 20.4 (Oct. 1978), pp. 801–

1978
[63]

Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later

Cleve Moler and Charles Van Loan. “Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later”. In: SIAM Review 45.1 (Jan. 2003), pp. 3–49. issn: 0036-1445, 1095-7200. doi: 10 . 1137 / S00361445024180. url: http://epubs.siam.org/doi/10.1137/S00361445...

2003 doi
[64]

A Chebychev propagator with iterative time order- ing for explicitly time-dependent Hamiltonians

Mamadou Ndong et al. “A Chebychev propagator with iterative time order- ing for explicitly time-dependent Hamiltonians”. In:The Journal of Chemical Physics 132.6 (Feb. 14, 2010), p. 064105. issn: 0021-9606, 1089-7690. doi: 10.1063/1.3312531 . url: http://aip.scitation.org/doi/...

2010 doi
[65]

A Simplex Method for Function Minimization

J. A. Nelder and R. Mead. “A Simplex Method for Function Minimization”. In: The Computer Journal 7.4 (1965), pp. 308,313. issn: 0010-4620. 59

1965
[66]

Michael A. Nielsen. Quantum computation and quantum information . Cam- bridge: Cambridge University Press, 2000. isbn: 0-521-63235-8

2000
[67]

Unsupervised Machine Learning on a Hybrid Quantum Computer

J. S. Otterbach et al. “Unsupervised Machine Learning on a Hybrid Quantum Computer”. In: ArXiv e-prints (Dec. 2017)

2017
[68]

Web graph similarity for anomaly detection

Panagiotis Papadimitriou, Ali Dasdan, and Hector Garcia-Molina. “Web graph similarity for anomaly detection”. In: Journal of Internet Services and Applications 1.1 (May 2010), pp. 19–30. issn: 1869-0238. doi: 10.1007/ s13174-010-0003-x . url: https://doi.org/10.1007/s13174-010-0003- x

2010 doi
[69]

Breaking the 49-Qubit Barrier in the Simulation of Quan- tum Circuits

E. Pednault et al. “Breaking the 49-Qubit Barrier in the Simulation of Quan- tum Circuits”. In: ArXiv e-prints (Oct. 2017)

2017
[70]

Direct search algorithms for optimization calculations

M. J. D. Powell. “Direct search algorithms for optimization calculations”. In: Acta Numerica 7 (1998), pp. 287–336. doi: 10.1017/S0962492900002841

1998 doi
[71]

RASCAL: Cal- culation of Graph Similarity using Maximum Common Edge Subgraphs

John W. Raymond, Eleanor J. Gardiner, and Peter Willett. “RASCAL: Cal- culation of Graph Similarity using Maximum Common Edge Subgraphs”. In: The Computer Journal 45.6 (2002), pp. 631,644. issn: 0010-4620

2002
[72]

Stochastic global optimization methods part II: Multi level methods

A. Rinnooy Kan and G. Timmer. “Stochastic global optimization methods part II: Multi level methods”. In: Mathematical Programming 39.1 (1987), pp. 57,78. issn: 0025-5610

1987
[73]

Functional stability analysis of numerical algorithms

Thomas H. Rowan. “Functional stability analysis of numerical algorithms”. PhD thesis. 1990. 218 pp. url: https://search-proquest-com.ezproxy. library.uwa.edu.au/docview/303865032?accountid=14681

1990
[74]

Theory of Quantum Annealing of an Ising Spin Glass

G. E. Santoro. “Theory of Quantum Annealing of an Ising Spin Glass”. In: Science 295.5564 (Mar. 29, 2002), pp. 2427–2430. issn: 00368075, 10959203. doi: 10.1126/science.1068774. url: http://www.sciencemag.org/cgi/ doi/10.1126/science.1068774 (visited on 10/15/2018)

2002 doi
[75]

Permutation Generation Methods

Robert Sedgewick. “Permutation Generation Methods”. In: ACM Comput. Surv. 9.2 (June 1977), pp. 137–164. issn: 0360-0300. doi: 10.1145/356689. 356692. url: http://doi.acm.org/10.1145/356689.356692

1977
[76]

A functional architecture for scalable quantum computing

E. A. Sete, W. J. Zeng, and C. T. Rigetti. “A functional architecture for scalable quantum computing”. In: 2016 IEEE International Conference on Rebooting Computing (ICRC) . Oct. 2016, pp. 1–6. doi: 10 . 1109 / ICRC . 2016.7738703

2016
[77]

Polynomial-time algorithms for prime factorization and dis- crete logarithms on a quantum computer

Peter W Shor. “Polynomial-time algorithms for prime factorization and dis- crete logarithms on a quantum computer”. In: SIAM journal on computing 5 (1997), SIAM journal on computing , (5), p.1509. issn: 0097-5397. 60

1997
[78]

Algorithms for quantum computation: discrete logarithms and factoring

P.W. Shor. “Algorithms for quantum computation: discrete logarithms and factoring”. In: Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium on (1994), pp. 124,134

1994
[79]

Expokit: a software package for computing matrix expo- nentials

Roger B. Sidje. “Expokit: a software package for computing matrix expo- nentials”. In: ACM Transactions on Mathematical Software 24.1 (Mar. 1, 1998), pp. 130–156. issn: 00983500. doi: 10.1145/285861.285868 . url: http://portal.acm.org/citation.cfm?doid=285861.285868 (visited o...

1998
[80]

Mastering the game of Go with deep neural networks and tree search

David Silver et al. “Mastering the game of Go with deep neural networks and tree search”. In: Nature 529.7587 (2016), pp. 484–9. issn: 0028-0836

2016
[81]

qHiPSTER: The Quantum High Performance Software Testing Environment

M. Smelyanskiy, N. P. D. Sawaya, and A. Aspuru-Guzik. “qHiPSTER: The Quantum High Performance Software Testing Environment”. In: ArXiv e- prints (Jan. 2016)

2016
[82]

Automatic design quality evaluation using graph similarity mea- sures

B. Strug. “Automatic design quality evaluation using graph similarity mea- sures”. In: Automation in Construction 32 (2013), pp. 187 –195. issn: 0926-

2013
[83]

Quantum waveguide theory: A direct solu- tion to the time-dependent Schr¨ odinger equation

J. B. Wang and S. Midgley. “Quantum waveguide theory: A direct solu- tion to the time-dependent Schr¨ odinger equation”. In: Physical Review B 60.19 (Nov. 15, 1999), pp. 13668–13675. issn: 0163-1829, 1095-3795. doi: 10.1103/PhysRevB.60.13668 . url: https://link.aps.org/doi/10....

1999 doi
[84]

Time-dependent approach to scattering by Chebyshev-polynomial expansion and the fast-Fourier-transform algorithm

J. B. Wang and T. T. Scholz. “Time-dependent approach to scattering by Chebyshev-polynomial expansion and the fast-Fourier-transform algorithm”. In: Physical Review A 57.5 (May 1, 1998), pp. 3554–3559. issn: 1050-2947, 1094-1622. doi: 10.1103/PhysRevA.57.3554 . url: https://li...

1998 doi
[85]

Training a quan- tum optimizer

Dave Wecker, Matthew B. Hastings, and Matthias Troyer. “Training a quan- tum optimizer”. In: Phys. Rev. A 94.2 (Aug. 2016), p. 022309. doi: 10 . 1103/PhysRevA.94.022309 . url: https://link.aps.org/doi/10.1103/ PhysRevA.94.022309

2016
[86]

Weisstein

Eric W. Weisstein. Graph Isomorphism Complete . MathWorld - A Wolfram Web Resource. 2018.url: http://mathworld.wolfram.com/GraphIsomorphismComplete. html (visited on 09/13/2018). 61

2018
[87]

Efficient quantum circuits for diagonal unitaries with- out ancillas

Jonathan Welch et al. “Efficient quantum circuits for diagonal unitaries with- out ancillas”. In: New Journal of Physics 16.3 (Mar. 31, 2014), p. 033040. issn: 1367-2630. doi: 10 . 1088 / 1367 - 2630 / 16 / 3 / 033040. url: http : / / stacks . iop . org / 1367 - 2630 / 16 / i ...

2014
[88]

Face recognition by elastic bunch graph matching

L. Wiskott et al. “Face recognition by elastic bunch graph matching”. In: IEEE Transactions on Pattern Analysis and Machine Intelligence 19.7 (July 1997), pp. 775–779. issn: 0162-8828. doi: 10.1109/34.598235

1997 doi
[89]

Graph similarity scoring and matching

Laura A. Zager and George C. Verghese. “Graph similarity scoring and matching”. In: Applied Mathematics Letters 21.1 (2008), pp. 86 –94. issn: 0893-9659. doi: https://doi.org/10.1016/j.aml.2007.01.006 . url: http://www.sciencedirect.com/science/article/pii/S0893965907001012

2008 doi
[90]

Efficient Graph Similarity Search Over Large Graph Databases

Weiguo Zheng et al. “Efficient Graph Similarity Search Over Large Graph Databases”. In: Knowledge and Data Engineering, IEEE Transactions on 27.4 (2015), pp. 964,978. issn: 1041-4347. 62 APPENDIX A Original Honours Proposal Title: Quantum Graph Similarity and Applications Auth...

2015
[97]

The QAOA can be formulated to reduce a combinatorial optimisation problem to a parameter search on around two variables

S(G1, G2) = S(G2, G1) 63 Aim To investigate a quantum algorithmic approach to graph similarity and its applica- tions this project will examine graph similarity by applying the recently proposed ’Quantum Approximate Optimisation Algorithm’ (QAOA) [29]. The QAOA can be formulat...

2018
[173]

isbn: 0-387-95014-1

New York: Springer, 2000. isbn: 0-387-95014-1

2000
[836]

issn: 0036-1445, 1095-7200. doi: 10 . 1137 / 1020098. url: http : / / epubs.siam.org/doi/10.1137/1020098 (visited on 09/17/2018)

2018 doi
[1201]

doi: 10.1137/S0895479899356080

issn: 0895-4798, 1095-7162. doi: 10.1137/S0895479899356080. url: http://epubs.siam.org/doi/10.1137/S0895479899356080 (visited on 08/11/2018)

2018 doi
[1506]

url: http://doi.wiley.com/10.1002/%28SICI%291099- 1506%28200001/02%297%3A1%3C27%3A%3AAID- NLA185%3E3.0.CO%3B2- 4 (visited on 09/17/2018)

doi: 10.1002/(SICI)1099-1506(200001/02)7:1<27::AID-NLA185> 3.0.CO;2- 4. url: http://doi.wiley.com/10.1002/%28SICI%291099- 1506%28200001/02%297%3A1%3C27%3A%3AAID- NLA185%3E3.0.CO%3B2- 4 (visited on 09/17/2018)

2018 doi
[2018]

url: https://research.googleblog.com/2018/03/a-preview-of- bristlecone-googles-new.html

2018
[5805]

org / 10

doi: https : / / doi . org / 10 . 1016 / j . autcon . 2012 . 12 . 015. url: http://www.sciencedirect.com/science/article/pii/S0926580512002543

2012

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Simulation of Many-Body Fermi Sys- tems on a Universal Quantum Computer

Daniel S. Abrams and Seth Lloyd. “Simulation of Many-Body Fermi Sys- tems on a Universal Quantum Computer”. In: Physical Review Letters 79.13 (Sept. 29, 1997), pp. 2586–2589. issn: 0031-9007, 1079-7114. doi: 10.1103/ PhysRevLett . 79 . 2586. url: https : / / link . aps . org / doi / 10 . 1103 / PhysRevLett.79.2586 (visited on 10/15/2018)

1997

[2] [18]

Graph comparison via nonlinear quantum search

M. Chiew et al. “Graph comparison via nonlinear quantum search”. In: arXiv:1810.01647 [quant-ph] (Oct. 3, 2018). arXiv: 1810.01647. url: http: //arxiv.org/abs/1810.01647 (visited on 10/15/2018)

arXiv 2018

[3] [2]

Quantum attacks on Bitcoin, and how to protect against them

D. Aggarwal et al. “Quantum attacks on Bitcoin, and how to protect against them”. In: ArXiv e-prints (Oct. 2017)

2017

[4] [3]

A New Scaling and Squar- ing Algorithm for the Matrix Exponential

Awad H. Al-Mohy and Nicholas J. Higham. “A New Scaling and Squar- ing Algorithm for the Matrix Exponential”. In: SIAM Journal on Matrix Analysis and Applications 31.3 (Jan. 2010), pp. 970–989. issn: 0895-4798, 1095-7162. doi: 10.1137/09074721X. url: http://epubs.siam.org/doi/ 10.1137/09074721X (visited on 09/16/2018)

doi:10.1137/09074721x 2010

[5] [4]

Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators

Awad H. Al-Mohy and Nicholas J. Higham. “Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators”. In: SIAM Journal on Scientific Computing 33.2 (Jan. 2011), pp. 488–511. issn: 1064-8275, 1095-7197. doi: 10.1137/100788860. url: http://epubs.siam. org/doi/10.1137/100788860 (visited on 08/02/2018)

doi:10.1137/100788860 2011

[6] [5]

Matrix exponentials and parallel prefix computation in a quantum control problem

T. Auckenthaler et al. “Matrix exponentials and parallel prefix computation in a quantum control problem”. In: Parallel Computing 36.5 (June 2010), pp. 359–369. issn: 01678191. doi: 10.1016/j.parco.2010.01.006 . url: http://linkinghub.elsevier.com/retrieve/pii/S0167819110000244 (visited on 09/17/2018)

work page doi:10.1016/j.parco.2010.01.006 2010

[7] [6]

PETSc Web page

Satish Balay et al. PETSc Web page . 2018. url: http://www.mcs.anl. gov/petsc

2018

[8] [7]

Beating the random assignment on constraint satisfaction problems of bounded degree

Boaz Barak et al. “Beating the random assignment on constraint satisfaction problems of bounded degree”. In: arXiv:1505.03424 [cs] (May 13, 2015). arXiv: 1505.03424. url: http://arxiv.org/abs/1505.03424 (visited on 10/05/2018)

arXiv 2015