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Marius Drechsler
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#import "@preview/fletcher:0.5.1": diagram, node, edge
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#import "@preview/gentle-clues:0.9.0": example
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#import "@preview/quill:0.3.0": quantum-circuit, lstick, rstick, ctrl, targ, mqgate, meter
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= Background
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== Quantum Computation and Quantum Circuits
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A quantum computer is a device that performs calculations by using certain phenomena of quantum mechanics.
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The algorithms that run on this device are specified in quantum circuits.
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#example[
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@example_qc shows a simple quantum circuit that implements a specific quantum algorithm.
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#figure(
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quantum-circuit(
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lstick($|0〉$), $H$, mqgate($U$, n: 2, width: 5em, inputs: ((qubit: 0, label: $x$), (qubit: 1, label: $y$)), outputs: ((qubit: 0, label: $x$), (qubit: 1, label: $y plus.circle f(x)$))), $H$, meter(), [\ ],
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lstick($|1〉$), $H$, 1, 1, 1
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),
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caption: [A quantum circuit implementing the Deutsch-Jozsa algorithm]
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) <example_qc>
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]
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== Decision Diagrams
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Decision diagrams in general are directed acyclical graphs, that may be used to express control flow through a series of conditions.
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It consists of a set of decision nodes and terminal nodes.
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The decision nodes represent an arbitrary decision based on an input value and may thus have any number of outgoing edges.
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The terminal nodes represent output values and may not have outgoing edges.
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A @bdd is a specific kind of decision diagram, where there are two terminal nodes (0 and 1) and each decision node has two outgoing edges, depending solely on a single bit of an input value.
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@bdd[s] may be used to represent any boolean function.
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#example[
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Example @bdd[s] implementing boolean functions with an arity of $2$ are show in @example_bdd_xor and @example_bdd_and.
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#figure(
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diagram(
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node-stroke: .1em,
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node((0, 0), [$x_0$], radius: 1em),
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edge((0, 0), (-1, 1), [0], "->"),
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edge((0, 0), (1, 1), [1], "->"),
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node((-1, 1), [$x_1$], radius: 1em),
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node((1, 1), [$x_1$], radius: 1em),
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edge((-1, 1), (-1, 2), [0], "->"),
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edge((-1, 1), (1, 2), [1], "->"),
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edge((1, 1), (1, 2), [0], "->"),
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edge((1, 1), (-1, 2), [1], "->"),
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node((-1, 2), [$0$]),
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node((1, 2), [$1$]),
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),
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caption: [A @bdd for an XOR gate.]
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) <example_bdd_xor>
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#figure(
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diagram(
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node-stroke: .1em,
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node((1, 0), [$x_0$], radius: 1em),
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edge((1, 0), (0, 2), [0], "->"),
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edge((1, 0), (1, 1), [1], "->"),
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node((1, 1), [$x_1$], radius: 1em),
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edge((1, 1), (0, 2), [0], "->"),
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edge((1, 1), (1, 2), [1], "->"),
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node((0, 2), [$0$]),
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node((1, 2), [$1$]),
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),
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caption: [A @bdd for an AND gate.]
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) <example_bdd_and>
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]
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#import "@preview/tablex:0.0.8": tablex
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#import "@preview/unify:0.6.0": qty
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= Benchmarks
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== Google Benchmark
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== MQT QCEC Bench
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To generate test cases for the application schemes, @mqt Bench was used. @quetschlich2023mqtbench
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#tablex(
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columns: (1fr, 1fr, 1fr),
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rows: (auto, auto, auto),
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[*Benchmark Name*], [*Diff Run Time*], [*Proportional Run Time*],
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[DJ], [$qty("1.2e-6", "s")$], [$qty("1.5e-6", "s")$],
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[Grover], [$qty("1.3e-3", "s")$], [$qty("1.7e-3", "s")$]
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)
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#import "@preview/glossarium:0.4.1": *
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= Comparison of @smhdt and @bach
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During the course of this work, we took a look at two distinct helper-data algorithms: the S-Metric Helper Data method and the newly presented method of optimization through Boundary Adaptive Clustering with helper-data.
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The S-Metric method will always outperform BACH considering the amount of helper data needed for operation.
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This comes from the nature of S-Metric quickly approaching an optimal @ber for a certain bit width and not improving any further for higher amounts of metrics.
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Comparing both formulas for the extracted-bits to helper-data-bits ratio for both methods we can quickly see that S-Metric will always yield more extracted bits per helper-data bit than BACH.
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Considering #glspl("ber"), S-Metric does outperform BACH for smaller symbol widths.
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But while the error rate for higher order quantization rises exponentially for higher-order bit quantizations, the #glspl("ber") of BACH do seem to rise rather linear than exponentially for higher-order bit quantizations.
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This behaviour might be attributed to the general procedure of shaping the input values for the quantizer in such a way that they are clustered around the center of a quantizer step, which is a property that carries on for higher order bit quantizations.
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We can now compare both the S-Metric Helper Data method and the newly presented method of optimization through Boundary Adaptive Clustering with Helper data based on the @ber and the amount of helper data bits, more specifically the extracted bit to helper data bit ratio $cal(r)$.
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The ratios $cal(r)$ for both methods are defined as:
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$
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cal(r)_"SMHD" = frac(M, log_2(S))\
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cal(r)_"BACH" = frac(M, N-1)
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$
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A good outline to compare both performances of @bach and @smhdt is if $cal(r) = 1$ for varying values of $M$.
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#figure(
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table(
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columns: 7,
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[*M*], [$1$], [$2$], [$3$], [$4$], [$5$], [$6$],
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[*@ber @smhdt*], [$8 dot 10^(-6)$], [$4 dot 10^(-5)$], [$1 dot 10^(-3)$], [$0.02$], [$0.08$], [$0.15$],
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[*@ber @bach*], [$0.09$], [$0.05$], [$0.05$], [$0.22$], [$0.23$], [$0.23$]
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),
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caption: [#glspl("ber") for @bach and @smhdt configurations that equal the bit width of the quantized symbol and the amount of helper data bits]
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)
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#import "@preview/lovelace:0.3.0": pseudocode-list
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= Implementation
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== Visualisation
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Initially, a visualisation of the diff algorithms applied to quantum circuits was created to assess their usefulness in equivalence checking.
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Additionally, this served as exercise to better understand the algorithms to be used for the implementation in @qcec.
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== QCEC Application Scheme
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The Myers' Algorithm was implemented as an application scheme in @qcec.
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#figure(
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block(
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pseudocode-list[
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+ do something
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+ do something else
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+ *while* still something to do
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+ do even more
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+ *if* not done yet *then*
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+ wait a bit
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+ resume working
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+ *else*
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+ go home
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+ *end*
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+ *end*
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],
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width: 100%
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),
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caption: [Myers' algorithm.]
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) <myers_algorithm>
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== QCEC Benchmarking Tool
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As @qcec doesn’t have built-in benchmarks, a benchmarking tool was developed to test different configurations on various circuit pairs.
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= State of the Art
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There are a variety of existing approaches to providing a suitable oracle for quantum circuit equivalence checking based on @dd[s].
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@qcec currently implements gate-cost, lookahead, one-to-one, proportional and sequential application schemes. @burgholzer2021ec
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