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arxiv: 2201.10574 · v10 · submitted 2022-01-25 · 🪐 quant-ph · cs.CC

Basic Quantum Algorithms

Renato Portugal This is my paper

Pith reviewed 2026-05-24 12:20 UTC · model grok-4.3

classification 🪐 quant-ph cs.CC
keywords quantum algorithmscircuit modelDeutsch algorithmShor algorithmGrover algorithmphase estimationHHL algorithmquantum computing foundations
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The pith

A review supplies circuit-model descriptions of the foundational quantum algorithms from Deutsch in 1985 to HHL in 2009.

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

The paper revisits the sequence of early quantum algorithms that began with Deutsch's 1985 method for evaluating a function at two points at once. It then covers Deutsch-Jozsa for distinguishing constant and balanced functions, Bernstein-Vazirani for identifying linear Boolean functions, Simon's algorithm for detecting one-to-one versus two-to-one functions, Shor's algorithms for factoring and discrete logarithms, Kitaev's phase estimation, Grover's search, and the HHL algorithm for linear systems. Emphasis falls on faithful re-presentations in the circuit model without added technical content. A sympathetic reader would care because these examples continue to anchor understanding of quantum speedups as the field updates its foundations.

Core claim

The paper claims that the earliest quantum algorithms can be described in detail using the circuit model, thereby revisiting and rewriting the foundations of the theory as the field evolves rapidly.

What carries the argument

The circuit model, which encodes quantum algorithms as sequences of gates on qubits.

If this is right

  • Deutsch's algorithm demonstrates simultaneous evaluation via superposition.
  • Simon's algorithm achieves an exponential speedup over classical methods for its problem.
  • Shor's algorithms solve integer factoring and discrete logarithms efficiently enough to threaten classical cryptography.
  • Grover's algorithm delivers a quadratic speedup for unstructured search.
  • The HHL algorithm solves systems of linear equations with a logarithmic dependence on problem size.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A common circuit presentation could make it easier to spot structural similarities when designing new algorithms.
  • As quantum hardware matures, these same circuit descriptions may need minor adjustments to account for error models not present in the original papers.
  • The review format could support direct comparisons of query complexity between quantum and classical versions of each problem.

Load-bearing premise

The paper assumes that the listed algorithms from Deutsch through HHL form the complete set of foundational examples worth revisiting and that their original formulations can be re-presented in circuit language without introducing new technical content.

What would settle it

An explicit demonstration that any one of the circuit diagrams deviates from the original algorithm's specification or adds new technical content would challenge the faithful re-presentation.

Figures

Figures reproduced from arXiv: 2201.10574 by Renato Portugal.

Figure 2.1
Figure 2.1. Figure 2.1: Bloch sphere and the location of states |0i, |1i, |±i, and |±ii. An arbitrary state |ψi is shown with spherical angles θ and ϕ [PITH_FULL_IMAGE:figures/full_fig_p010_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Histogram of the probability distribution generated by measuring a qubit whose [PITH_FULL_IMAGE:figures/full_fig_p013_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Example of a circuit with a Hadamard gate and a meter (Reprint Courtesy of IBM [PITH_FULL_IMAGE:figures/full_fig_p015_2_3.png] view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: Output of the circuit with a Hadamard gate and a meter (Reprint Courtesy of IBM [PITH_FULL_IMAGE:figures/full_fig_p016_2_4.png] view at source ↗
Figure 2.5
Figure 2.5. Figure 2.5: Circuit with gates H and CNOT (Reprint Courtesy of IBM Corporation ©) [PITH_FULL_IMAGE:figures/full_fig_p026_2_5.png] view at source ↗
Figure 2.6
Figure 2.6. Figure 2.6: Output of the circuit of Fig [PITH_FULL_IMAGE:figures/full_fig_p026_2_6.png] view at source ↗
Figure 2.7
Figure 2.7. Figure 2.7: Basic bricks to build a Boolean expression. To implement the AND and OR gates, [PITH_FULL_IMAGE:figures/full_fig_p028_2_7.png] view at source ↗
Figure 2.8
Figure 2.8. Figure 2.8: Simplification rules for Boolean expressions. [PITH_FULL_IMAGE:figures/full_fig_p029_2_8.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Truth tables of all 1-bit Boolean functions. [PITH_FULL_IMAGE:figures/full_fig_p033_3_1.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Truth tables of all 2-bit balanced functions. [PITH_FULL_IMAGE:figures/full_fig_p045_4_1.png] view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: Oracle of Simon’s algorithm. The only trivial simplification that can be immediately seen is that the first two multiqubit Toffoli gates can be simplified into only one Toffoli gate with empty control on qubit 2, full control on qubit 3 and target on qubit 4 [PITH_FULL_IMAGE:figures/full_fig_p061_6_1.png] view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: Probability distribution p(`) as a function of ` given by Eq. (7.2) when N = 21 (q = 29 , a = 2, r = 6, c = 85). The dots at the top the peaks correspond to ` = 0, 85, 171, 256, 341, 427. When we look at p(`) as a continuous function, we are able to show that it is a truly periodic function, while the function depicted in [PITH_FULL_IMAGE:figures/full_fig_p072_7_1.png] view at source ↗
Figure 7.2
Figure 7.2. Figure 7.2: (a) Numerator of p(`) as a continuous function of ` for ` ∈ [0, q/r] when N = 21 (q = 29 , r = 6, c = 85). (b) Denominator of p(`) as a continuous function of ` without qc. (c) p(`) as a continuous function of ` for ` ∈ [0, q/r]. Plot (c) is obtained by dividing (a) by (b) by qc. The denominator of p(`) (without qc), sin2 (π`r/q), is also the square of a sinusoidal function, which is zero only at ` = 0 a… view at source ↗
Figure 7.3
Figure 7.3. Figure 7.3: The upper part of the envelope of p(`) as a function of ` in the domain [0, q] when N = 21 (q = 29 , r = 6, c = 85), which is obtained using the square of the cosecant function (csc x = 1/ sin x). There is an alternative route to obtain the shape of p(`) in the domain [0, q]. The numerator of p(`), sin2 π`rc q , is the square of a sinusoidal function quickly oscillating inside an envelope, the lower part… view at source ↗
Figure 7.4
Figure 7.4. Figure 7.4: First peak of the probability distribution [PITH_FULL_IMAGE:figures/full_fig_p074_7_4.png] view at source ↗
Figure 7.5
Figure 7.5. Figure 7.5: Snapshot of a Jupyter notebook showing the decomposition of the Fourier transform [PITH_FULL_IMAGE:figures/full_fig_p080_7_5.png] view at source ↗
Figure 8.1
Figure 8.1. Figure 8.1: Lattice with basis r = (16, 0) and s = (−11, 1) modulo 16 and cyclic boundaries. As an example of application to the calculation of discrete logarithms, consider again N = 34 but this time take a = 27 and G27 = {27, 15, 31, 21, 23, 9, 5, 33, 7, 19, 3, 13, 11, 25, 29, 1}. Suppose we want to calculate log27 3, whose answer we know: s = 11. Since the order of a = 27 is r = 16 modulo 34, the lattice L with b… view at source ↗
Figure 8.2
Figure 8.2. Figure 8.2: Circuit of the discrete logarithm algorithm when [PITH_FULL_IMAGE:figures/full_fig_p087_8_2.png] view at source ↗
Figure 9.1
Figure 9.1. Figure 9.1: Depiction of vectors |x0i and |di. The angle θ is very small for large N, and in this case, θ/2 is a good approximation of sin(θ/2). In addition, the sine of an angle is equal to the cosine of the complement, that is, θ 2 ≈ sin θ 2 = cos  π 2 − θ 2  . As (π − θ)/2 is the angle between |x0i and |di, by definition of the inner product, cos (π − θ)/2 is the inner product of vectors |x0i and |di, the resul… view at source ↗
Figure 9.2
Figure 9.2. Figure 9.2: Vector uf |di is a reflexion of |di about the horizontal axis. The next step is to apply g = [PITH_FULL_IMAGE:figures/full_fig_p097_9_2.png] view at source ↗
Figure 9.3
Figure 9.3. Figure 9.3: Vector g uf |di is a reflexion of uf |di about |di [PITH_FULL_IMAGE:figures/full_fig_p097_9_3.png] view at source ↗
Figure 9.4
Figure 9.4. Figure 9.4: Vector |ψi is the final state before measurement and a is the norm of the projection of |x0i on |ψi. The angle between |ψi and |x0i is less than or equal to θ/2. Vector |ψi is almost orthogonal to |di at this point, as depicted in [PITH_FULL_IMAGE:figures/full_fig_p098_9_4.png] view at source ↗
Figure 10.1
Figure 10.1. Figure 10.1: The first block of the phase estimation circuit is made of [PITH_FULL_IMAGE:figures/full_fig_p102_10_1.png] view at source ↗
Figure 10.2
Figure 10.2. Figure 10.2: Full circuit of the QPE algorithm. Algorithm 10.1: Quantum phase estimation algorithm Input: Eigenvector |ψi of U. Output: Number φ˜, where exp(2πiφ) is the eigenvalue of |ψi and φ˜ ≈ φ2 m. 1 Prepare the initial state |0i ⊗m ⊗ |ψi; 2 Apply H⊗m to the first register; 3 For ` in [0, m − 1] apply the controlled operation C m−`  U 2 `  , where the control qubit is m − ` and the target is the 2nd register;… view at source ↗
Figure 10.3
Figure 10.3. Figure 10.3: Quantum part of Shor’s algorithm based on QPE. [PITH_FULL_IMAGE:figures/full_fig_p106_10_3.png] view at source ↗
Figure 10.4
Figure 10.4. Figure 10.4: Circuit of the discrete logarithm algorithm based on QPE, where [PITH_FULL_IMAGE:figures/full_fig_p108_10_4.png] view at source ↗
read the original abstract

Quantum computing is evolving so rapidly that it forces us to revisit, rewrite, and update the foundations of the theory. \emph{Basic Quantum Algorithms} revisits the earliest quantum algorithms. The journey began in 1985 with Deutsch attempting to evaluate a function at two domain points simultaneously. Then, in 1992, Deutsch and Jozsa created a quantum algorithm that determines whether a Boolean function is constant or balanced. The following year, Bernstein and Vazirani realized that essentially the same algorithm could be used to identify a specific Boolean function within a set of linear Boolean functions. In 1994, Simon introduced a novel quantum algorithm that determines whether a function is one-to-one or two-to-one exponentially faster than any classical algorithm for the same problem. That same year, Shor developed two groundbreaking quantum algorithms for integer factoring and calculating discrete logarithms, posing a threat to widely used cryptographic methods. In 1995, Kitaev proposed an alternative formulation based on phase estimation that proved valuable in numerous applications. The following year, Grover devised a quantum search algorithm that is quadratically faster than its classical counterpart. More than a decade later, Harrow, Hassidim, and Lloyd proposed a quantum algorithm for solving systems of linear equations, now known as the HHL algorithm. With an emphasis on the circuit model, this work provides a detailed description of all these remarkable algorithms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript is an expository survey that revisits eight foundational quantum algorithms (Deutsch 1985, Deutsch-Jozsa 1992, Bernstein-Vazirani 1993, Simon 1994, Shor 1994 for factoring and discrete log, Kitaev 1995 phase estimation, Grover 1996, and HHL 2009), providing detailed descriptions with emphasis on their quantum circuit model implementations.

Significance. If the circuit descriptions faithfully reproduce the action and resource counts of the cited originals without introducing errors, the work could serve as a consolidated pedagogical reference for the circuit-model presentation of these algorithms. However, because the manuscript introduces no new algorithms, proofs, or technical results, its significance is limited to exposition and does not advance the research frontier.

minor comments (3)
  1. [Abstract] Abstract: the phrase 'all these remarkable algorithms' is vague; explicitly listing the eight algorithms and stating the precise scope (earliest foundational examples only) would improve clarity.
  2. [Introduction] Introduction: the opening claim that quantum computing 'forces us to revisit, rewrite, and update the foundations' lacks supporting citations to recent developments that would justify a new survey at this time.
  3. Throughout: ensure every algorithm section states the exact oracle assumptions, query complexity, and gate counts so readers can directly compare with the original papers without external lookup.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and the recommendation of minor revision. The manuscript is explicitly positioned as an expository survey providing consolidated circuit-model descriptions of foundational algorithms. We address the referee's observations on significance below.

read point-by-point responses

  1. Referee: because the manuscript introduces no new algorithms, proofs, or technical results, its significance is limited to exposition and does not advance the research frontier.

    Authors: We agree that the work contains no new algorithmic results or proofs. Its intended contribution is pedagogical: to supply a single, self-contained reference with explicit circuit diagrams, gate counts, and step-by-step derivations that faithfully reproduce the original papers. In a field that evolves rapidly, such updated and unified expositions can reduce the barrier for students and researchers who must otherwise consult multiple primary sources. We therefore view the manuscript as a survey whose value lies precisely in its expository completeness rather than in novel technical claims. revision: no

Circularity Check

0 steps flagged

No significant circularity; expository survey of known algorithms

full rationale

The paper is a survey that revisits and describes eight established quantum algorithms (Deutsch 1985 through HHL 2009) in circuit-model language. Its central claim is faithful re-presentation of these prior results, with no original derivations, predictions, fitted parameters, or new mathematical content asserted. All load-bearing steps trace to external citations whose correctness can be verified independently of this manuscript; no self-definitional reductions, fitted-input predictions, or self-citation chains appear in the derivation chain because no derivation chain exists.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a review paper containing no new mathematical claims, derivations, or postulates.

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