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Understanding Quantum Information and Computation

T0 review · 0 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This course claims that the full theory of quantum computing—from single-qubit state vectors through the core algorithms to density matrices, quantum channels, and fault-tolerant error correction—can be learned in one self-contained…

desk verdict Free, well-built quantum computing course; nothing scientifically new, but the teaching craft and precision are real, and the visible math checks out. read the letter →

arxiv 2507.11536 v1 pith:CDGISVWF submitted 2025-07-15 quant-ph

John Watrous This is my paper · ORCID
classification quant-ph MSC 81P6881P45 PACS 03.67.-a03.67.Lx
keywords quantumcomputinginformationcircuitsentanglementdensitymatriceschannelserrorcorrectionfaulttolerance
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 course sets out to demonstrate that the theory of quantum computing can be learned from first principles in a single, self-contained sequence of sixteen lessons. The sequence begins with quantum states as complex unit vectors and builds up through quantum circuits, the core algorithms, the density-matrix and channel formalism, and finally quantum error correction and fault tolerance. Its central claim is that this full arc is accessible to a reader with basic linear algebra, and that the delicate ideas—entanglement, measurement, and noise—can be presented through concrete mathematical objects rather than hand-waving. If the course succeeds on its own terms, it offers a rigorous public route into quantum information for learners who do not have access to university courses.

What carries the argument

The load-bearing object is the quantum circuit model together with the simplified formulation of quantum information, in which states are unit vectors, operations are unitary matrices, and measurements are standard-basis or projective rules. This pair provides the derivational engine for the early lessons: every protocol is reduced to multiplying matrices and reading squared amplitudes. The later lessons replace the simplified formulation with density matrices and quantum channels precisely because the simplified version cannot describe the reduced state of one subsystem, and that replacement is what makes quantum error correction treatable. A second key object is the e-bit, the unit of entanglement embodied by the shared state $|\phi^+\rangle = (|00\rangle + |11\rangle)/\sqrt2$, which the course uses to make teleportation, superdense coding, and the CHSH strategy concrete.

What would settle it

A direct experimental check would be to prepare many copies of the plus state $|+\rangle = (|0\rangle + |1\rangle)/\sqrt2$, measure each in the standard basis, and test whether the outcome frequencies converge to $1/2$ each; a systematic deviation would contradict the Born rule on which the course's measurement analyses are built.

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Extended reading notes

Core claim

On its own terms, the work claims that the whole of quantum information and computation admits a coherent pedagogical derivation from a small set of linear-algebraic structures: Dirac notation, tensor products, unitary operations, and projective measurements, later generalized to density matrices, quantum channels, and general measurements. The course argues that this simplified formulation is already enough to analyze teleportation, superdense coding, the CHSH game, the no-cloning theorem, and the impossibility of perfectly discriminating non-orthogonal states; the general formulation then supplies the tools needed for reduced states, noisy evolution, and error correction. Entanglement is treated as a concrete resource measured in e-bits, with the shared two-qubit state $|\phi^+\rangle = (|00\rangle + |11\rangle)/\sqrt2$ as the unit that powers the protocols. The final claim is that the stabilizer formalism and the toric code turn abstract error correction into an explicit path to fault-tolerant computation.

Load-bearing premise

The course rests on the standard measurement postulate that when a quantum system is measured, each outcome appears with probability equal to the squared amplitude and the system is left in the corresponding projected state; every protocol and algorithm analysis in the course inherits this assumption.

Editorial extensions

If this is right

  • A reader who masters the first four lessons can derive teleportation and superdense coding directly from unitary circuits and the shared entangled state, rather than taking them as black-box facts.
  • The no-cloning theorem and the impossibility of perfect discrimination of non-orthogonal states follow from linearity and inner products, so the same foundations that enable quantum protocols also delimit what they cannot do.
  • The CHSH analysis shows that a quantum strategy wins with probability $(2+\sqrt2)/4 \approx 0.85$, above the classical maximum of $3/4$, giving a concrete operational test of entanglement.
  • Once density matrices and channels are introduced, noise can be described within the same formalism, and the stabilizer formalism plus the toric code show how error correction can be made fault-tolerant.

Reading between the lines

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

  • A consequence the text leaves implicit: its ordering, with projective measurements before general measurements, could be tested against a density-matrix-first curriculum by comparing learners' ability to analyze teleportation and noisy states.
  • If the course's accessibility claim is right, then the main barrier to entering quantum computing is mathematical maturity rather than access to specialized teaching, which would make a self-contained open course a meaningful educational intervention; this is an inference, not something the text itself tests.
  • The CHSH game could serve as a compact diagnostic for whether a learner has genuinely understood entanglement: because the quantum advantage appears as a single number that lies strictly between the classical maximum and impossibility, a learner who can reproduce the derivation has mastered the relevant linear algebra.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. This manuscript is a textbook-style course on the theory of quantum computing, consisting of a preface, a table of contents for 16 lessons in four units, and the full text of Lessons 1-4 (Single Systems, Multiple Systems, Quantum Circuits, Entanglement in Action). It develops the formalism of quantum states, measurements, unitary operations, tensor products, and entanglement, with worked proofs of the no-cloning theorem, the impossibility of perfect discrimination of non-orthogonal states, and detailed analyses of teleportation, superdense coding, and the CHSH game. The material is presented as an educational exposition with no new research claims; the measurement postulates are stated explicitly, and standard results such as Tsirelson's inequality are attributed to their originators.

Significance. Within the visible portion, the mathematical exposition is careful and internally consistent: the four-branch teleportation analysis, the Bell-state encoding of superdense coding, and the case-by-case CHSH calculation all check out against the stated definitions, and the no-cloning and state-discrimination proofs are valid. The course's explicit treatment of the Born rule and projective collapse as postulates, its transparent ordering conventions, and its worked derivations make it a potentially valuable rigorous introduction for advanced undergraduates and beginning graduate students. The adoption of the Born rule and projective collapse as postulates is explicit and is not a defect for a course at this level. If Lessons 5-16 maintain this standard, the full course would be a significant freely available educational resource. The manuscript makes no empirical or mathematical claims beyond established results, so its significance is pedagogical rather than research-oriented.

minor comments (6)
  1. [§4.3 (Set-up)] The Set-up paragraph says 'together the two qubits (X, Y) are in the |ϕ+⟩ state'; this should refer to (A, B), since Alice's and Bob's qubits were named A and B in the preceding sentence.
  2. [§3.2] In the paragraph on extending orthonormal sets to bases, the sentence 'The last n − m columns ban be filled' contains a typo; it should read 'can be filled'.
  3. [§3.3] The proof of the impossibility of perfect discrimination is presented for circuits consisting of unitary gates followed by a single standard-basis measurement. Since the section states the result in general terms, it would be helpful to note explicitly that this restriction is without loss of generality, citing the implementation of projective measurements in §3.2 and the standard possibility of deferring intermediate measurements.
  4. [§4.3] Tsirelson's bound is stated as the optimal quantum winning probability without proof or a visible citation. Please ensure the bibliography includes a precise reference, and consider adding a sentence indicating that the proof is beyond the scope of the lesson.
  5. [§4.3] The historical claim that Tsirelson 'first described the CHSH experiment as a game' would benefit from a citation; if no citation is available, consider softening the wording.
  6. [§4.1] The argument ruling out qubit transmission by classical communication alone is presented as an intuitive no-cloning argument. The text already notes that a formal proof uses quantum information theory; it may be worth flagging explicitly that the no-cloning argument is a heuristic account rather than a complete proof, so that readers do not mistake it for the full argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the course is an explicit pedagogical exposition with no fitted parameters, and its worked derivations follow from stated postulates rather than from conclusions it aims to establish.

full rationale

The document makes no original scientific claim and fits no parameters, so the standard circularity patterns do not arise. The Born rule is introduced as an explicit postulate in Lesson 1.2 ('if a quantum state is measured, each classical state of the system appears with probability equal to the absolute value squared of the entry ... This is known as the Born rule in quantum mechanics'), and projective collapse is similarly introduced as a measurement rule in Lesson 3.2; these are pedagogical assumptions, not hidden inputs disguised as outputs. The teleportation, superdense-coding, and CHSH analyses are direct algebraic consequences of the stated unitary-evolution and measurement rules, and the CHSH case-by-case probabilities are computed from the constructed rotations rather than imported as a conclusion. The no-cloning and non-orthogonal-discrimination arguments are proved in-line from linearity and unitarity. Tsirelson's inequality and Holevo's theorem are attributed to their original authors and are not used to justify any claim that then re-derives them from the course's own framework. The only self-reference is the preface's description of this text as a 'Director's Cut' of the author's IBM Quantum Learning course; that statement concerns provenance, not a load-bearing mathematical premise. Consequently there is no step in which a prediction or derivation reduces to its own inputs, and the appropriate circularity score is 0.

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

The course fits no free parameters to data. It depends on the standard postulates of quantum mechanics and on standard linear algebra, which are listed as axioms. No new physical entities are introduced; Alice, Bob, e-bits, and Bell states are standard pedagogical devices, not novel postulates.

assumptions (5)
  • domain assumption Quantum states are unit vectors in a complex inner product space, and standard basis measurement probabilities are squared absolute values of amplitudes.
    Adopted as the basic framework in Lesson 1.2 and used throughout the course.
  • domain assumption Closed-system evolution is represented by unitary matrices.
    Introduced in Lesson 1.2 and used for circuits, algorithms, and protocols.
  • domain assumption Measurements are described by projection operators with Born-rule probabilities and post-measurement collapse.
    Defined in Lesson 3.2 and used in partial measurement and teleportation analyses.
  • domain assumption No-signaling: measuring one system cannot change the outcome probabilities of a distant system.
    Invoked in Lesson 2.1 to justify reduced probabilistic states and in Lesson 2.2 for partial quantum measurements.
  • standard math Standard linear algebra facts about tensor products, inner products, and orthonormal bases are valid.
    Used throughout Lessons 1 through 4 to derive no-cloning, teleportation, and the CHSH strategy.

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Cite this review

Pith. "Pith review of Understanding Quantum Information and Computation." pith.science (2026). https://pith.science/paper/CDGISVWF

@misc{pith2026250711536,
  author       = {Pith},
  title        = {Pith review of: Understanding Quantum Information and Computation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CDGISVWF}},
  note         = {Machine review of arXiv:2507.11536}
}
read the original abstract

This is a course on the theory of quantum computing. It consists of 16 lessons, each with a video and written component, covering the basics of quantum information, quantum algorithms (including query algorithms, Shor's algorithm for integer factorization, and Grover's algorithm), the general formulation of quantum information (including density matrices, quantum channels, and general measurements), and quantum error correction (including the basics, the stabilizer formalism, CSS codes, the toric code, and fault-tolerant quantum computation).

Figures

Figures reproduced from arXiv: 2507.11536 by the authors.

Figure 3.1
Figure 3.1. A Boolean circuit for computing the exclusive-OR of two bits. [PITH_FULL_IMAGE:figures/full_fig_p070_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. The same Boolean circuit as in Figure [PITH_FULL_IMAGE:figures/full_fig_p071_3_2.png] view at source ↗
Figure 3.3
Figure 3.3. The same Boolean circuit as in Figure [PITH_FULL_IMAGE:figures/full_fig_p072_3_3.png] view at source ↗
Figures from the paper (143 more)
Figure 3.4
Figure 3.4. Figure 3.4: depicts an arithmetic circuit that takes two variable input values (x and y) as well as a third input set to the value 1. The values carried by the wires, as functions of the values x and y, are shown in the figure. x y 1 ∗ + + ∗ x x y y x 2 y + 1 x 2 + y x 2y + x 2 …
Figure 3.5
Figure 3.5. Figure 3.5: A simple quantum circuit on one qubit. Sometimes we may wish to explicitly indicate the input or output states of circuits. For example, if we apply the operation THSH to the state |0⟩, we obtain the state 1+i 2 |0⟩ + √ 1 2 |1⟩. This can be indicated as is shown in …
Figure 3.6
Figure 3.6. Figure 3.6: The circuit from Figure [PITH_FULL_IMAGE:figures/full_fig_p073_3_6.png]
Figure 3.7
Figure 3.7. Figure 3.7: A simple quantum circuit on two qubits. Before examining this circuit in greater detail and explaining what it does, it is imperative that we first clarify how qubits are ordered in quantum circuits. This connects with the convention that Qiskit uses for naming and o…
Figure 3.8
Figure 3.8. Figure 3.8: The action of a controlled-NOT gate on standard basis states. [PITH_FULL_IMAGE:figures/full_fig_p076_3_8.png]
Figure 3.9
Figure 3.9. Figure 3.9: A quantum circuit including measurements and classical bit wires. [PITH_FULL_IMAGE:figures/full_fig_p077_3_9.png]
Figure 3.10
Figure 3.10. Figure 3.10: A compact representation of the circuit in Figure [PITH_FULL_IMAGE:figures/full_fig_p077_3_10.png]
Figure 3.11
Figure 3.11. Figure 3.11: Six common single-qubit gates. + [PITH_FULL_IMAGE:figures/full_fig_p078_3_11.png]
Figure 3.13
Figure 3.13. Figure 3.13: A swap gate. As the course continues, we’ll see more examples of quantum circuits, which are usually more complicated than the simple examples above. Here are some examples of symbols used to denote gates that commonly appear in circuit diagrams. • Single-qubit gate…
Figure 3.14
Figure 3.14. Figure 3.14: A controlled-NOT gate, a controlled-controlled NOT (or Toffoli) gate, [PITH_FULL_IMAGE:figures/full_fig_p079_3_14.png]
Figure 3.15
Figure 3.15. Figure 3.15: A unitary operation U as a quantum gate along with a controlled version of this gate. 3.2 Inner products and projections To better prepare ourselves to explore the capabilities and limitations of quantum circuits, we’ll now introduce some additional mathematical con…
Figure 3.16
Figure 3.16. Figure 3.16: A quantum circuit for copying a standard basis state. [PITH_FULL_IMAGE:figures/full_fig_p095_3_16.png]
Figure 3.17
Figure 3.17. Figure 3.17: A quantum circuit U perfectly discriminates the states |ψ⟩ and |ϕ⟩. for |ψ⟩ and 1 for |ϕ⟩; the analysis would not differ fundamentally if these output values were reversed. Notice that, in addition to the qubits that initially store either |ψ⟩ or |ϕ⟩, the circuit is…
Figure 4.1
Figure 4.1. Figure 4.1: The quantum teleportation protocol expressed as a quantum circuit. [PITH_FULL_IMAGE:figures/full_fig_p103_4_1.png]
Figure 4.2
Figure 4.2. Figure 4.2: Three states |π0⟩, |π1⟩, and |π2⟩ relevant to the analysis of the teleporta￾tion protocol. Analysis To analyze the teleportation protocol, we’ll examine the behavior of the circuit described above, one step at a time, beginning with the situation in which Q is initia…
Figure 4.3
Figure 4.3. Figure 4.3: depicts the same circuit as before, with the addition of the system R (represented by a collection of qubits on the top of the diagram that nothing happens to). X Z α|0⟩|γ0⟩ + β|1⟩|γ1⟩    |ϕ +⟩    H + Alice Bob |π0⟩ |π1⟩ |π2⟩ [PITH_FULL_IMAGE:figur…
Figure 4.4
Figure 4.4. Figure 4.4: The superdense coding protocol described as a quantum circuit. [PITH_FULL_IMAGE:figures/full_fig_p112_4_4.png]
Figure 4.5
Figure 4.5. Figure 4.5: The interactions between the Referee and Alice and Bob in a nonlocal [PITH_FULL_IMAGE:figures/full_fig_p115_4_5.png]
Figure 4.6
Figure 4.6. Figure 4.6: A quantum strategy in which Alice and Bob make use of a shared [PITH_FULL_IMAGE:figures/full_fig_p118_4_6.png]
Figure 4.7
Figure 4.7. Figure 4.7: A quantum circuit description of Alice and Bob’s strategy. [PITH_FULL_IMAGE:figures/full_fig_p121_4_7.png]
Figure 4.8
Figure 4.8. Figure 4.8: Alice’s basis is determined by the angle [PITH_FULL_IMAGE:figures/full_fig_p125_4_8.png]
Figure 4.9
Figure 4.9. Figure 4.9: Bob’s basis is determined by the angle β. Now, if we combine together (4.3) and (4.4) we get the formula ⟨ψα ⊗ ψβ|ϕ +⟩ = 1 √ 2 ⟨ψα|ψβ⟩, which works for all real numbers α and β [PITH_FULL_IMAGE:figures/full_fig_p125_4_9.png]
Figure 4.10
Figure 4.10. Figure 4.10: Alice and Bob’s bases when x = 0 and y = 0. Following the same sort of analysis that we went through above, but with α and β being variables, we find this [PITH_FULL_IMAGE:figures/full_fig_p126_4_10.png]
Figure 4.11
Figure 4.11. Figure 4.11: Alice and Bob’s bases when x = 0 and y = 1. while the probability they disagree is the sine-squared of this angle, sin2 π 8  = 2 − √ 2 4 . When (x, y) = (0, 1), Alice and Bob choose α = 0 and β = −π/8, resulting in the bases shown in [PITH_FULL_IMAGE:figures/full…
Figure 4.12
Figure 4.12. Figure 4.12: Alice and Bob’s bases when x = 1 and y = 0. |ψπ/4⟩ |ψ3π/4⟩ |ψ−π/8⟩ |ψ3π/8⟩ [PITH_FULL_IMAGE:figures/full_fig_p128_4_12.png]
Figure 4.13
Figure 4.13. Figure 4.13: Alice and Bob’s bases when x = 1 and y = 1. When (x, y) = (1, 1), Alice and Bob choose α = π/4 and β = −π/8. This results in the bases shown in [PITH_FULL_IMAGE:figures/full_fig_p128_4_13.png]
Figure 5.1
Figure 5.1. Figure 5.1: A simple abstraction of a standard model of computation. [PITH_FULL_IMAGE:figures/full_fig_p134_5_1.png]
Figure 5.2
Figure 5.2. Figure 5.2: An abstraction of the query model of computation [PITH_FULL_IMAGE:figures/full_fig_p135_5_2.png]
Figure 5.3
Figure 5.3. Figure 5.3: A classical query gate. When a Boolean circuit is created for a query problem, the input function f is accessed through these gates, and the number of queries that the circuit makes is simply the number of query gates that appear in the circuit. The input wires of th…
Figure 5.4
Figure 5.4. Figure 5.4: A Boolean circuit that solves the parity problem for a function [PITH_FULL_IMAGE:figures/full_fig_p138_5_4.png]
Figure 5.5
Figure 5.5. Figure 5.5: The action of a unitary query gate [PITH_FULL_IMAGE:figures/full_fig_p138_5_5.png]
Figure 5.6
Figure 5.6. Figure 5.6: describes Deutsch’s algorithm as a quantum circuit. |0⟩ |1⟩ H H Uf H ( 0 f is constant 1 f is balanced [PITH_FULL_IMAGE:figures/full_fig_p140_5_6.png]
Figure 5.7
Figure 5.7. Figure 5.7: Three states |π1⟩, |π2⟩, and |π3⟩ considered in the analysis of Deutsch’s algorithm. The initial state is |1⟩|0⟩, and the two Hadamard operations on the left-hand side of the circuit transform this state to |π1⟩ = |−⟩|+⟩ = 1 2 [PITH_FULL_IMAGE:figures/full_fig_p141_…
Figure 5.8
Figure 5.8. Figure 5.8: The Deutsch–Jozsa algorithm as a quantum circuit. [PITH_FULL_IMAGE:figures/full_fig_p144_5_8.png]
Figure 5.9
Figure 5.9. Figure 5.9: The quantum circuit portion of Simon’s algorithm. [PITH_FULL_IMAGE:figures/full_fig_p155_5_9.png]
Figure 6.1
Figure 6.1. Figure 6.1: A simple abstraction of a standard model of computation. [PITH_FULL_IMAGE:figures/full_fig_p166_6_1.png]
Figure 6.2
Figure 6.2. Figure 6.2: The standard basis measurement on the left can be deferred through [PITH_FULL_IMAGE:figures/full_fig_p171_6_2.png]
Figure 6.3
Figure 6.3. Figure 6.3: A Boolean circuit for computing the exclusive-OR of two bits. [PITH_FULL_IMAGE:figures/full_fig_p172_6_3.png]
Figure 6.4
Figure 6.4. Figure 6.4: A Boolean circuit implementing a half adder using two FANOUT gates, [PITH_FULL_IMAGE:figures/full_fig_p176_6_4.png]
Figure 6.5
Figure 6.5. Figure 6.5: A full adder constructed from two half adders and an OR gate. [PITH_FULL_IMAGE:figures/full_fig_p176_6_5.png]
Figure 6.6
Figure 6.6. Figure 6.6: Cascading a half adder and three full adders creates a Boolean circuit for [PITH_FULL_IMAGE:figures/full_fig_p177_6_6.png]
Figure 6.7
Figure 6.7. Figure 6.7: The action of a Toffoli gate on a standard basis state. [PITH_FULL_IMAGE:figures/full_fig_p183_6_7.png]
Figure 6.8
Figure 6.8. Figure 6.8: A quantum circuit implementation of a Toffoli gate. [PITH_FULL_IMAGE:figures/full_fig_p184_6_8.png]
Figure 6.9
Figure 6.9. Figure 6.9: Implementations of AND, OR, and FANOUT gates using Toffoli and [PITH_FULL_IMAGE:figures/full_fig_p185_6_9.png]
Figure 6.10
Figure 6.10. Figure 6.10: depicts the actions of the circuits C and R side-by-side. C t gates x       f(x) R O(t) gates |x⟩    |0 k ⟩       | f(x)⟩    |g(x)⟩ [PITH_FULL_IMAGE:figures/full_fig_p186_6_10.png]
Figure 6.11
Figure 6.11. Figure 6.11: An implementation of a SWAP gate using three CNOT gates. [PITH_FULL_IMAGE:figures/full_fig_p187_6_11.png]
Figure 6.12
Figure 6.12. Figure 6.12: A garbage-free implementation of the original Boolean circuit [PITH_FULL_IMAGE:figures/full_fig_p188_6_12.png]
Figure 6.13
Figure 6.13. Figure 6.13: The circuit in Figure [PITH_FULL_IMAGE:figures/full_fig_p188_6_13.png]
Figure 6.14
Figure 6.14. Figure 6.14: A unitary implementation of an invertible function [PITH_FULL_IMAGE:figures/full_fig_p190_6_14.png]
Figure 7.1
Figure 7.1. Figure 7.1: A unitary operation U (viewed as a quantum gate) on the left and a controlled-U operation on the right. We can create a quantum circuit for a controlled-U operation by first adding a control qubit to the circuit for U, and then replacing every gate in the circuit for…
Figure 7.2
Figure 7.2. Figure 7.2: A phase estimation circuit with a single control qubit. [PITH_FULL_IMAGE:figures/full_fig_p197_7_2.png]
Figure 7.3
Figure 7.3. Figure 7.3: The states |π0⟩, . . . , |π3⟩ considered in the analysis of the single control qubit phase estimation procedure. Next, the controlled-U operation is performed, which results in the state |π2⟩ = 1 √ 2 |ψ⟩|0⟩ + 1 √ 2 [PITH_FULL_IMAGE:figures/full_fig_p198_7_3.png]
Figure 7.4
Figure 7.4. Figure 7.4: Output probabilities for phase estimation with a single control qubit. [PITH_FULL_IMAGE:figures/full_fig_p199_7_4.png]
Figure 7.5
Figure 7.5. Figure 7.5: A modified version of the circuit in Figure [PITH_FULL_IMAGE:figures/full_fig_p200_7_5.png]
Figure 7.6
Figure 7.6. Figure 7.6: Output probabilities for phase estimation with a single control qubit [PITH_FULL_IMAGE:figures/full_fig_p201_7_6.png]
Figure 7.7
Figure 7.7. Figure 7.7: The initial portion of a quantum circuit for phase estimation with two [PITH_FULL_IMAGE:figures/full_fig_p202_7_7.png]
Figure 7.8
Figure 7.8. Figure 7.8: The states |π0⟩, . . . , |π3⟩ considered in the analysis of two-qubit phase estimation. this: |π3⟩ = |ψ⟩ ⊗ 1 2 1 ∑ a0=0 1 ∑ a1=0 e 2πi(2a1+a0)θ |a1a0⟩. If we think about the binary string a1a0 as representing an integer x ∈ {0, 1, 2, 3} in binary notation, which is x…
Figure 7.9
Figure 7.9. Figure 7.9: The complete quantum circuit for phase estimation with two control [PITH_FULL_IMAGE:figures/full_fig_p205_7_9.png]
Figure 7.10
Figure 7.10. Figure 7.10: Output probabilities for phase estimation with two control qubits. [PITH_FULL_IMAGE:figures/full_fig_p206_7_10.png]
Figure 7.11
Figure 7.11. Figure 7.11: Three equivalent ways to denote controlled-phase gates. [PITH_FULL_IMAGE:figures/full_fig_p211_7_11.png]
Figure 7.12
Figure 7.12. Figure 7.12: A quantum circuit for performing the operation ( [PITH_FULL_IMAGE:figures/full_fig_p211_7_12.png]
Figure 7.13
Figure 7.13. Figure 7.13: A quantum circuit for QFT32 using an operation for QFT16. Analysis The key formula we need to verify that the circuit just described implements the 2 m-dimensional quantum Fourier transform is this one: (−1) abω xy 2m−1ω ay 2m = ω (2x+a)(2 m−1 b+y) 2m . This formula…
Figure 7.14
Figure 7.14. Figure 7.14: A quantum circuit for the general phase estimation procedure. [PITH_FULL_IMAGE:figures/full_fig_p216_7_14.png]
Figure 7.15
Figure 7.15. Figure 7.15: Arc and chord lengths on the complex unit circle. [PITH_FULL_IMAGE:figures/full_fig_p218_7_15.png]
Figure 7.16
Figure 7.16. Figure 7.16: Output probabilities for the outcomes 3, 4, and 5 in the phase estimation [PITH_FULL_IMAGE:figures/full_fig_p221_7_16.png]
Figure 7.17
Figure 7.17. Figure 7.17: Output probabilities for the outcomes 7, 8, and 9 in the phase estimation [PITH_FULL_IMAGE:figures/full_fig_p221_7_17.png]
Figure 8.1
Figure 8.1. Figure 8.1: An implementation of a phase query gate Zf using a standard query gate Uf . requires that one workspace qubit, initialized to a |−⟩ state, is made available. This qubit remains in the |−⟩ state after the implementation has completed, and can be reused (to implement s…
Figure 8.2
Figure 8.2. Figure 8.2: A quantum circuit implementation of the Grover operation on [PITH_FULL_IMAGE:figures/full_fig_p242_8_2.png]
Figure 8.3
Figure 8.3. Figure 8.3: A quantum circuit running Grover’s algorithm for [PITH_FULL_IMAGE:figures/full_fig_p243_8_3.png]
Figure 8.4
Figure 8.4. Figure 8.4: The action of Zf , which reflects about the line L1, on a vector |ψ⟩ that is a real linear combination of |A0⟩ and |A1⟩. L2 |u⟩ |ψ⟩ H⊗nZORH⊗n |ψ⟩ |A0⟩ |A1⟩ [PITH_FULL_IMAGE:figures/full_fig_p250_8_4.png]
Figure 8.5
Figure 8.5. Figure 8.5: The action of H⊗nZORH⊗n , which reflects about the line L2, on a vec￾tor |ψ⟩ that is a real linear combination of |A0⟩ and |A1⟩ [PITH_FULL_IMAGE:figures/full_fig_p250_8_5.png]
Figure 8.6
Figure 8.6. Figure 8.6: The Grover operation G is a composition of the reflections about the lines L1 and L2. Its action on real linear combinations of |A0⟩ and |A1⟩ is to rotate by twice the angle between L1 and L2. 8.4 Choosing the number of iterations We have established that the state v…
Figure 9.1
Figure 9.1. Figure 9.1: Illustration of the Cartesian coordinates of a point on the unit [PITH_FULL_IMAGE:figures/full_fig_p277_9_1.png]
Figure 9.2
Figure 9.2. Figure 9.2: The states |0⟩, |1⟩, |+⟩, |−⟩, |+i⟩, and |−i⟩ on the Bloch sphere. Every point on the sphere can be described in this way — which is to say that the points we obtain when we range over all possible pure states of a qubit correspond precisely to a sphere in 3 real dim…
Figure 9.3
Figure 9.3. Figure 9.3: Qubit states of the form |ψα⟩ = cos(α)|0⟩ + sin(α)|1⟩ on the Bloch sphere. Here’s another class of quantum state vectors that has appeared from time to time throughout this course, including previously in this lesson. |ψα⟩ = cos(α)|0⟩ + sin(α)|1⟩ (for α ∈ [0, π)) The…
Figure 9.4
Figure 9.4. Figure 9.4: An illustration of the density matrix 1 2 |0⟩⟨0| + 1 2 |+⟩⟨+| inside the Bloch sphere. Multiple systems Density matrices can represent states of multiple systems in an analogous way to state vectors in the simplified formulation of quantum information, following the …
Figure 10.1
Figure 10.1. Figure 10.1: depicts such an implementation, in the form of a circuit diagram, for a channel whose input and output systems are the same system, X. In this diagram, the wires represent arbitrary systems, as indicated by the labels above the wires, and not necessarily single qubi…
Figure 10.2
Figure 10.2. Figure 10.2: shows an implementation of a channel Φ whose input system is X and whose output system is Y. This time the unitary operation transforms (W, X) into a pair (G, Y), where G is a new “garbage” system that gets traced out, leaving Y as the output system. U X W Y G ρ Φ(ρ…
Figure 10.3
Figure 10.3. Figure 10.3: A Stinespring representation of the completely dephasing channel. [PITH_FULL_IMAGE:figures/full_fig_p305_10_3.png]
Figure 10.4
Figure 10.4. Figure 10.4: An alternative Stinespring representation of the completely dephasing [PITH_FULL_IMAGE:figures/full_fig_p308_10_4.png]
Figure 10.5
Figure 10.5. Figure 10.5: A Stinespring representation of the qubit reset channel. [PITH_FULL_IMAGE:figures/full_fig_p309_10_5.png]
Figure 10.6
Figure 10.6. Figure 10.6: An alternative representation of the qubit reset channel. [PITH_FULL_IMAGE:figures/full_fig_p309_10_6.png]
Figure 10.7
Figure 10.7. Figure 10.7: Evaluating a channel on one-half of the maximally entangled state [PITH_FULL_IMAGE:figures/full_fig_p315_10_7.png]
Figure 11.1
Figure 11.1. Figure 11.1: The tetrahedral states form the vertices of a regular tetrahedron in [PITH_FULL_IMAGE:figures/full_fig_p332_11_1.png]
Figure 11.2
Figure 11.2. Figure 11.2: The implementation of a general measurement using a workspace [PITH_FULL_IMAGE:figures/full_fig_p341_11_2.png]
Figure 12.1
Figure 12.1. Figure 12.1: An implementation of a measurement for the Hughston–Jozsa– [PITH_FULL_IMAGE:figures/full_fig_p368_12_1.png]
Figure 12.2
Figure 12.2. Figure 12.2: The horizontal line corresponding to the fidelity and the vertical line [PITH_FULL_IMAGE:figures/full_fig_p372_12_2.png]
Figure 13.1
Figure 13.1. Figure 13.1: The probability that two or three bits flip during transmission for the [PITH_FULL_IMAGE:figures/full_fig_p382_13_1.png]
Figure 13.2
Figure 13.2. Figure 13.2: An encoding circuit for the 3-bit repetition code. [PITH_FULL_IMAGE:figures/full_fig_p382_13_2.png]
Figure 13.3
Figure 13.3. Figure 13.3: An error detection circuit for the 3-bit repetition code. [PITH_FULL_IMAGE:figures/full_fig_p384_13_3.png]
Figure 13.4
Figure 13.4. Figure 13.4: If no errors occur, the error detection circuit results in the outcome [PITH_FULL_IMAGE:figures/full_fig_p384_13_4.png]
Figure 13.5
Figure 13.5. Figure 13.5: A single bit-flip error is detected by the [PITH_FULL_IMAGE:figures/full_fig_p385_13_5.png]
Figure 13.6
Figure 13.6. Figure 13.6: shows a modified version of the encoding circuit from above, which will now be able to protect against phase-flip errors. The modification is very simple: we simply apply a Hadamard gate to each qubit after performing the two controlled-NOT gates. + + H H H α|0⟩ + β…
Figure 13.7
Figure 13.7. Figure 13.7: An error detection circuit for the modified [PITH_FULL_IMAGE:figures/full_fig_p388_13_7.png]
Figure 13.8
Figure 13.8. Figure 13.8: A simplification of the error detection circuit for the modified [PITH_FULL_IMAGE:figures/full_fig_p388_13_8.png]
Figure 13.9
Figure 13.9. Figure 13.9: If no errors occur, the error detection circuit results in the outcome [PITH_FULL_IMAGE:figures/full_fig_p389_13_9.png]
Figure 13.10
Figure 13.10. Figure 13.10: A single phase-flip error is detected by the modified [PITH_FULL_IMAGE:figures/full_fig_p390_13_10.png]
Figure 13.11
Figure 13.11. Figure 13.11: An encoding circuit for the 9-qubit Shor code. [PITH_FULL_IMAGE:figures/full_fig_p392_13_11.png]
Figure 13.12
Figure 13.12. Figure 13.12: Relationships among X and CNOT gates. Z + = + Z Z + Z = + Z + Z Z = + Z [PITH_FULL_IMAGE:figures/full_fig_p393_13_12.png]
Figure 13.13
Figure 13.13. Figure 13.13: Relationships among Z and CNOT gates [PITH_FULL_IMAGE:figures/full_fig_p393_13_13.png]
Figure 13.14
Figure 13.14. Figure 13.14: A phase-flip error on one of the qubits of the 9-qubit Shor code. [PITH_FULL_IMAGE:figures/full_fig_p395_13_14.png]
Figure 13.15
Figure 13.15. Figure 13.15: A phase-flip error within the middle block, such as the one indicated [PITH_FULL_IMAGE:figures/full_fig_p395_13_15.png]
Figure 13.16
Figure 13.16. Figure 13.16: To detect phase-flip errors, we can decode the inner code, run the [PITH_FULL_IMAGE:figures/full_fig_p396_13_16.png]
Figure 13.17
Figure 13.17. Figure 13.17: A simplification of the circuit in Figure [PITH_FULL_IMAGE:figures/full_fig_p397_13_17.png]
Figure 13.18
Figure 13.18. Figure 13.18: A bit-flip error and a phase-flip error on the same qubit in the [PITH_FULL_IMAGE:figures/full_fig_p398_13_18.png]
Figure 13.19
Figure 13.19. Figure 13.19: An equivalent circuit to the one in Figure [PITH_FULL_IMAGE:figures/full_fig_p399_13_19.png]
Figure 13.20
Figure 13.20. Figure 13.20: A plot illustrating the break-even point for the 9-qubit Shor code. [PITH_FULL_IMAGE:figures/full_fig_p401_13_20.png]
Figure 14.1
Figure 14.1. Figure 14.1: A quantum circuit based on phase estimation for non-destructively [PITH_FULL_IMAGE:figures/full_fig_p413_14_1.png]
Figure 14.2
Figure 14.2. Figure 14.2: A quantum circuit performing 3-qubit Pauli observable P2 ⊗ P1 ⊗ P0 non-destructively on the top three qubits. eigenvalues +1 and −1, just like we usually have for phase estimation with one control qubit. Note that the control qubit is on the bottom in this diagram, …
Figure 14.3
Figure 14.3. Figure 14.3: A quantum circuit implementing a Z ⊗ Z measurement on the top two qubits. |0⟩ + + [PITH_FULL_IMAGE:figures/full_fig_p414_14_3.png]
Figure 14.4
Figure 14.4. Figure 14.4: A simplification of the circuit in Figure [PITH_FULL_IMAGE:figures/full_fig_p414_14_4.png]
Figure 14.5
Figure 14.5. Figure 14.5: A circuit that simultaneously measures the Pauli observables [PITH_FULL_IMAGE:figures/full_fig_p416_14_5.png]
Figure 14.6
Figure 14.6. Figure 14.6: The stabilizer generators Z ⊗ Z ⊗ I and I ⊗ Z ⊗ Z split the 8- dimensional space corresponding to three qubits into four 2-dimensional subspaces spanned by the standard basis states indicated in the squares corresponding to the measurement outcomes. two bits have od…
Figure 14.7
Figure 14.7. Figure 14.7: An encoding circuit for the 7-qubit Steane code. [PITH_FULL_IMAGE:figures/full_fig_p430_14_7.png]
Figure 14.8
Figure 14.8. Figure 14.8: Composing a lowest-weight correction operation [PITH_FULL_IMAGE:figures/full_fig_p435_14_8.png]
Figure 15.1
Figure 15.1. Figure 15.1: A 9 × 9 lattice. To realize this sort of configuration physically requires three dimensions. In particular, we could form the lattice into a cylinder by first matching up the left and right sides, and then bend the cylinder around so that the circles at the ends, wh…
Figure 15.2
Figure 15.2. Figure 15.2: A 9 × 9 lattice with periodic boundaries embedded on the surface of a torus. The way one can “move around” on a torus like this, between adjacent points on the lattice, will likely be familiar to those that have played old-school video games, where moving off the to…
Figure 15.3
Figure 15.3. Figure 15.3: Qubits, indicated by blue circles, are placed on the edges of the lattice. [PITH_FULL_IMAGE:figures/full_fig_p454_15_3.png]
Figure 15.4
Figure 15.4. Figure 15.4: The two types of stabilizer generators for the toric code. [PITH_FULL_IMAGE:figures/full_fig_p454_15_4.png]
Figure 15.5
Figure 15.5. Figure 15.5: Examples of stabilizer generators of the two types are indicated by [PITH_FULL_IMAGE:figures/full_fig_p455_15_5.png]
Figure 15.6
Figure 15.6. Figure 15.6: When an X and a Z stabilizer generator overlap, it is always on exactly two qubits — implying that the stabilizer generators commute. ones, and similarly, any one of the X stabilizer generators can be expressed as the product of the remaining X stabilizer generators…
Figure 15.7
Figure 15.7. Figure 15.7: The effect of a single X error on the Z stabilizer generator measurement outcomes. error therefore flips the parity of the four bits on both of the tiles it touches, causing the stabilizer generator measurements to output −1. Next let’s introduce multiple X errors t…
Figure 15.8
Figure 15.8. Figure 15.8: The effect of a chain of adjacent X errors on the Z stabilizer generator measurement outcomes. In such a case, an even number of X errors have occurred on every tile, so every stabilizer generator measurement results in a +1 outcome. Closed loops of adjacent X error…
Figure 15.9
Figure 15.9. Figure 15.9: A closed loop of adjacent X errors goes undetected by the toric code. unaffected qubit qubit affected by X error +1 measurement outcome −1 measurement outcome [PITH_FULL_IMAGE:figures/full_fig_p460_15_9.png]
Figure 15.10
Figure 15.10. Figure 15.10: The closed loop of adjacent X errors illustrated in [PITH_FULL_IMAGE:figures/full_fig_p460_15_10.png]
Figure 15.11
Figure 15.11. Figure 15.11: Such a chain of errors is not contained in the stabilizer because every X stabilizer generator places an even number of X operations on every horizontal line and every vertical line of qubits. This is therefore an actual example of a nontrivial error that the code …
Figure 15.12
Figure 15.12. Figure 15.12: A chain of adjacent X errors being corrected by an adjacent chain of X corrections. choosing a shortest path of X corrections between two −1 syndrome measurement outcomes, will properly correct the error that caused this syndrome. Perhaps more likely, depending on …
Figure 15.13
Figure 15.13. Figure 15.13: A chain of adjacent X corrections failing to correct a chain of adjacent X errors. unaffected qubit qubit corrected by X gate +1 measurement outcome −1 measurement outcome [PITH_FULL_IMAGE:figures/full_fig_p464_15_13.png]
Figure 15.14
Figure 15.14. Figure 15.14: Multiple X correction chains forming a minimum-weight perfect matching between −1 measurement outcomes [PITH_FULL_IMAGE:figures/full_fig_p464_15_14.png]
Figure 15.15
Figure 15.15. Figure 15.15: A surface code with smooth edges on the sides and rough edges on [PITH_FULL_IMAGE:figures/full_fig_p465_15_15.png]
Figure 15.16
Figure 15.16. Figure 15.16: A diagram of a rotated surface code. Black faces denote [PITH_FULL_IMAGE:figures/full_fig_p466_15_16.png]
Figure 15.17
Figure 15.17. Figure 15.17: A graphical representation of the 7-qubit Steane code. [PITH_FULL_IMAGE:figures/full_fig_p467_15_17.png]
Figure 15.18
Figure 15.18. Figure 15.18: A graphical representation of a [[19, 1, 5]] color code. Color codes are so-named because one of the required conditions on the graphs that define them is that the faces can be three-colored, meaning that the faces can each be assigned one of three colors in such a…
Figure 16.1
Figure 16.1. Figure 16.1: A teleportation circuit. are perfect. For example, if we decide to use a surface code for error correction, and a classical perfect matching algorithm is run to compute corrections, we really don’t need to concern ourselves with the possibility that errors in this c…
Figure 16.2
Figure 16.2. Figure 16.2: A fault-tolerant implementation of the circuit in Figure [PITH_FULL_IMAGE:figures/full_fig_p475_16_2.png]
Figure 16.3
Figure 16.3. Figure 16.3: CNOT gates propagate X and Z errors [PITH_FULL_IMAGE:figures/full_fig_p476_16_3.png]
Figure 16.4
Figure 16.4. Figure 16.4: Multiple CNOT gates can further propagate [PITH_FULL_IMAGE:figures/full_fig_p477_16_4.png]
Figure 16.5
Figure 16.5. Figure 16.5: A transversal implementation of a CNOT gate for CSS codes. [PITH_FULL_IMAGE:figures/full_fig_p478_16_5.png]
Figure 16.6
Figure 16.6. Figure 16.6: An implementation of a T gate using a magic state. To check that this circuit works correctly, we can first compute the action of the CNOT gate on the input. T|+⟩ ⊗ |ψ⟩ CNOT 7−→ 1 √ 2 |0⟩ ⊗ T|ψ⟩ + 1 + i 2 |1⟩ ⊗ T † |ψ⟩ [PITH_FULL_IMAGE:figures/full_fig_p481_16_6.png]
Figure 16.7
Figure 16.7. Figure 16.7: An implementation of a T gate on an encoded qubit using an encoded magic state. need encoded magic states. The gates in the original T-gate circuit are here replaced by gadgets, which we assume are fault-tolerant. This particular figure therefore suggests that we al…
Figure 16.8
Figure 16.8. Figure 16.8: Magic state distillation on encoded states. [PITH_FULL_IMAGE:figures/full_fig_p484_16_8.png]
Figure 16.9
Figure 16.9. Figure 16.9: A circuit for measuring a stabilizer generator of the form [PITH_FULL_IMAGE:figures/full_fig_p486_16_9.png]
Figure 16.10
Figure 16.10. Figure 16.10: Circuits for detecting X and Z errors using Steane error correction. do this, in the sense that the size of the noisy circuit required is on the order of N times some constant power of the logarithm of N. To state the theorem more formally requires being specific a…

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