Steven GellerQuantum Computing, End to End

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    Part I. Mental Model

    1. Computation Is Physical
    2. Classical Information, Gates, Circuits, and Reversibility
    3. Probability vs Amplitude
    4. Qubits Without Mysticism
    5. Measurement and What It Does Not Mean
    6. Interference as the Engine of Quantum Algorithms
    7. Entanglement Without Faster-Than-Light Myths
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    Part I. Mental Model · Chapter 3

    Probability vs Amplitude

    A quantum state is not a list of probabilities, and treating it as one is the fastest way to misread every algorithm in this book. This chapter draws the exact line between amplitude and probability, then shows with a single gate what information you destroy when you cross that line too early.

    Steven Geller
    In this chapter 7 sections

    Reader question. What information do amplitudes contain that a probability distribution discards?

    Amplitudes carry sign and complex phase as well as magnitude; probabilities retain only squared magnitudes, so states with identical immediate statistics can interfere differently after a later unitary.

    Scope and non-goals.
    • Amplitudes are treated as model variables rather than directly observable or stored classical answers.
    • Density matrices and the full taxonomy of classical and quantum uncertainty are deferred.
    Same probabilities, different amplitudes Two states measurement cannot tell apart alpha beta |+> |-> amplitudes: what gates transform 1/2 |+> |-> P(0) P(1) P(0) P(1) probabilities: what measurement returns
    Figure 3.1. Notice that the two distinct states on the left produce identical bars on the right. An immediate measurement cannot tell |+> from |->; yet the sign difference survives inside the state and decides what the next gate does.

    Two columns that look identical to a meter

    Compare |+> and |-> in the computational basis.

    A probability is a nonnegative number that directly says how likely an outcome is. An amplitude is a number; often complex; that contributes to a probability only after you take its squared magnitude. Amplitudes can be positive, negative, or complex, which means they can cancel each other or reinforce each other before measurement enters the story.

    That cancellation and reinforcement is the heart of every quantum algorithm. A randomized classical algorithm can spread probability mass across many outcomes, but probabilities only ever add. A quantum algorithm can arrange amplitudes so that paths leading to wrong answers cancel and paths leading to right answers survive. Measurement then samples from a distribution that interference has already shaped.

    One sentence is worth memorizing: amplitudes are the quantities quantum operations transform; probabilities are what measurement produces from them. Most confused claims about quantum computing violate this sentence somewhere.

    Evidence boundary. Measurement probabilities are squared magnitudes of amplitudes. [Michael A. Nielsen] [John Watrous]

    Squaring discards direction

    Map complex amplitudes to nonnegative probabilities and identify lost phase.

    A one-qubit state is written ∣ψ⟩=α∣0⟩+β∣1⟩\lvert\psi\rangle=\alpha\lvert0\rangle+\beta\lvert1\rangle, where α\alpha and β\beta are complex amplitudes. The state is normalized when ∣α∣2+∣β∣2=1\lvert\alpha\rvert^2+\lvert\beta\rvert^2=1; total probability must come to one, because measurement always returns something.

    Measurement in the computational basis follows the Born rule: the outcome 0 appears with probability ∣α∣2\lvert\alpha\rvert^2, and 1 with probability ∣β∣2\lvert\beta\rvert^2. Square first, then interpret; never the other order.

    Here is the subtlety the rest of the book exploits. Take α=1/2\alpha=1/\sqrt2 with β=1/2\beta=1/\sqrt2, and both outcomes have probability 1/21/2. Now flip the sign so β=−1/2\beta=-1/\sqrt2: the probabilities are still 1/21/2 each. The relative sign is invisible to an immediate measurement; but it is real, it is stored in the state, and later gates respond to it.

    Evidence boundary. Relative phase affects later interference even when immediate basis probabilities agree. [John Preskill] [Michael A. Nielsen]

    Hadamard recovers a relative sign

    Calculate deterministic opposite outputs from equal initial probabilities.

    Evidence boundary. Global phase does not change measurement predictions, whereas relative phase can. [John Watrous] [Michael A. Nielsen]

    B=(∣0⟩,∣1⟩),α,β∈C,pi=∣αi∣2,ϕ mod 2πB=(\lvert0\rangle,\lvert1\rangle),\quad \alpha,\beta\in\mathbb C,\quad p_i=\lvert\alpha_i\rvert^2,\quad \phi\bmod 2\pi(3.1)

    Notation contract: Basis order (|0>,|1>); amplitudes α,β∈ℂ; probabilities p_i=|α_i|²; phase angles modulo 2π.

    Combine first, square last

    Generalize the interference order of operations.

    Amplitudes behave like internal representation, not output. They are hidden state that affects later behavior but is never returned by the API; the API is measurement, and it returns samples. You would not judge a service by dumping its heap; do not judge a quantum algorithm by staring at its statevector.

    This is also why simulators mislead. A statevector simulator will happily print every amplitude, because your classical machine is expensively simulating the quantum state. Real hardware hands you no such thing: it returns measurement outcomes from the bases you chose, at the shot count you paid for. Designs that depend on "reading the statevector" are designs that only work on simulators, and the distinction decides what scales.

    A diagnostic for probability-only explanations

    Test whether a proposed description predicts the second experiment.

    Amplitude-versus-probability phase lab

    Acceptance contract for Amplitude-versus-probability phase lab
    FieldReader-visible record
    FormatInteractive Argand/statevector panel plus exact two-state calculation
    VerificationFixture confirms equal input probability vectors and opposite post-H distributions; UI never exposes amplitude as a measurement result.
    AvailabilitySource-embedded acceptance record; no separate download is claimed
    {
  "artifact": "Amplitude-versus-probability phase lab",
  "format": "Interactive Argand/statevector panel plus exact two-state calculation",
  "acceptance_test": "Fixture confirms equal input probability vectors and opposite post-H distributions; UI never exposes amplitude as a measurement result.",
  "publication_state": "source-embedded contract and worked fixture"
}

    Scope boundary

    • Amplitudes are treated as model variables rather than directly observable or stored classical answers.
    • Density matrices and the full taxonomy of classical and quantum uncertainty are deferred.

    Depth commitment. One exact counterexample, one Argand explanation, and one transfer exercise.

    Practice problem

    Construct two normalized one-qubit states with identical Z-basis probabilities and maximally different outcomes after a chosen gate.

    Deliverable
    Two statevectors, gate choice, and exact before/after probability tables.
    Pass condition
    A reference calculation verifies normalization, equality of initial distributions, and total variation distance one after the gate.

    Verification record

    Expected solution form. Symbolic derivation plus a numerical statevector fixture.

    Model answer. Choose |+> and |->. Both give (1/2,1/2) in Z. Applying H maps them to |0> and |1>, whose outcome distributions have total-variation distance one, the maximum possible separation.

    Model result and check. The fixture compares distributions before and after the submitted unitary and checks the stated maximum separation.

    Acceptance test. A reference calculation verifies normalization, equality of initial distributions, and total variation distance one after the gate.

    Provenance

    Sources and review

    1. Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press. 2010textbook
    2. John Watrous. The Theory of Quantum Information. Cambridge University Press / University of Waterloo. 2018textbook
    3. John Preskill. Lecture Notes for Physics 219: Quantum Computation. California Institute of Technology. 2018graduate lecture notes

    The load-bearing claims in the chapter are mapped inline to this registered source set. A citation supports only the bounded claim beside it.

    External review: Open.

    How review worksRevision historyReport or inspect errata

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    Prerequisites for this chapter

    1. Computation Is PhysicalPart I. Mental Model · Chapter 1

    Continue the thread

    1. Qubits Without MysticismPart I. Mental Model · Chapter 4
    2. Interference as the Engine of Quantum AlgorithmsPart I. Mental Model · Chapter 6
    3. Complex Numbers for Quantum ComputingPart II. Mathematical Core · Chapter 8

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