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Quantum Mechanics

Quantum Information and Computation

Quantum information studies what quantum systems can encode, communicate, compute, measure, and simulate. The central question is operational: given preparations, controls, measurements, and a finite supply of time and other resources, how well can a specified task be performed?

This volume connects states and channels to algorithms, communication protocols, sensors, simulators, and error-corrected devices. Start with What Is Quantum Information? for the conceptual introduction, or use the chapter map below to enter a particular subject.

Required background. Quantum states and density operators describe preparations; tensor products describe registers and subsystems; quantum channels and noise describe physical transformations. The mathematical preparation guide supplies the linear algebra and probability used along the way.

Choose a starting question

Section titled “Choose a starting question”

For a guided sequence across subjects, use the Quantum Information Roadmap.

Preparation, transformation, and observation

Section titled “Preparation, transformation, and observation”

A common experiment prepares a state ρx\rho_x with classical label xx, applies a channel Eλ\mathcal E_\lambda with control settings λ\lambda, and measures a POVM {My}\{M_y\}. Its outcome probabilities are

p(y∣x,λ)=Tr⁡[MyEλ(ρx)],My≥0,∑yMy=I.p(y\mid x,\lambda) =\operatorname{Tr}[M_y\mathcal E_\lambda(\rho_x)], \qquad M_y\geq0,\qquad \sum_y M_y=I.

The same expression describes a receiver guessing a message, the final measurement of a circuit, and a sensor estimating a parameter. It separates three possible sources of error: the prepared state, the transformation, and the readout. If later operations depend on the postmeasurement state, a quantum instrument supplies the state update as well as the outcome probability.

A common channel acting on all input states is an assumption about the experiment. A fixed initially uncorrelated environment gives such a channel; arbitrary preparation-dependent system–environment correlations need a larger model. Likewise, leakage and loss must be included in the output space or recorded as explicit outcomes. A normalized postselected output generally depends nonlinearly on its input, even though the unnormalized outcome operation is linear.

What counts as a useful resource depends on the allowed operations. Entanglement cannot be created from a separable state by local operations and classical communication, even when that classical communication is unlimited. A nonlocal quantum interaction changes that restriction. Resource theories make this dependence precise; they do not assign one universal usefulness score to every quantum state.

Anchor 1: nonorthogonal states carry limited one-shot evidence

Section titled “Anchor 1: nonorthogonal states carry limited one-shot evidence”

Suppose a sender chooses ∣0⟩|0\rangle or ∣+⟩=(∣0⟩+∣1⟩)/2|+\rangle=(|0\rangle+|1\rangle)/\sqrt2 with equal probability. A receiver must always guess which was sent. Measuring in the computational basis and guessing optimally succeeds with probability 3/43/4: outcome 11 identifies ∣+⟩|+\rangle, while outcome 00 favors ∣0⟩|0\rangle.

The best quantum measurement does better, but cannot be perfect. With ρ0=∣0⟩⟨0∣\rho_0=|0\rangle\langle0| and ρ1=∣+⟩⟨+∣\rho_1=|+\rangle\langle+|, the Holevo–Helstrom bound gives

Psuccopt=12+14∥ρ0−ρ1∥1=12(1+12)≃0.8536.\begin{aligned} P_{\mathrm{succ}}^{\mathrm{opt}} &=\frac12+\frac14\|\rho_0-\rho_1\|_1\\ &=\frac12\left(1+\frac1{\sqrt2}\right) \simeq0.8536. \end{aligned}

Here ∥A∥1\|A\|_1 is the sum of the singular values. The optimal measurement separates the positive and negative eigenspaces of ρ0−ρ1\rho_0-\rho_1; Watrous, Chapter 3, develops the general result. This is a limit for one copy, equal priors, and minimum-error guessing without an inconclusive outcome. Changing those conditions changes the task.

From qubits to useful operations

Section titled “From qubits to useful operations”

A physical qubit is a selected two-level degree of freedom in a larger device. A logical qubit is an encoding, often involving many physical systems, whose errors are corrected or otherwise suppressed. The distinction matters because a device with many physical qubits may still have too much noise, too little connectivity, or too little execution time for a particular logical computation.

Consider preparing

∣GHZn⟩=∣0⟩⊗n+∣1⟩⊗n2.|\mathrm{GHZ}_n\rangle =\frac{|0\rangle^{\otimes n}+|1\rangle^{\otimes n}}{\sqrt2}.

A Hadamard followed by CNOT gates can spread entanglement from one qubit to the others. On a line, starting at an endpoint and adding the next site uses n−1n-1 CNOT layers. Starting near the center can improve the constant by growing in both directions, but bounded-range unitary gates still require depth proportional to the line’s length. With all-to-all connectivity and disjoint simultaneous gates, a doubling construction uses ⌈log⁡2n⌉\lceil\log_2 n\rceil entangling layers and n−1n-1 CNOTs. These statements concern unitary preparation from a product state; measurement and communication change the available constructions.

Gate count and gate depth are therefore different resources. Moreover, an operation that copies the two basis labels coherently creates α∣0⋯0⟩+β∣1⋯1⟩\alpha|0\cdots0\rangle+\beta|1\cdots1\rangle; it does not produce independent copies of α∣0⟩+β∣1⟩\alpha|0\rangle+\beta|1\rangle.

Comparisons of algorithms and devices also need the same output accuracy and treatment of preparation, sampling, classical processing, and retries. Reading quantum-technology claims explains how a theorem, a finite experiment, and an engineering projection support different conclusions. Research in quantum computing follows changing assessments.

Map of the volume

Section titled “Map of the volume”
Chapter What you will learn
Information-theoretic foundations Carriers, quantum entropy, distinguishability, entanglement measures, and resources.
Gates, circuits, and computation models Gates, universality, coherent control, measurements, and alternative computation models.
Algorithms and complexity How quantum algorithms work, what access they assume, and which speedups are proved.
Noise, channels, and mitigation How errors change outputs and when an estimator can compensate for them.
Error correction and fault tolerance Encodings, syndromes, decoders, logical operations, and resource requirements.
Communication, cryptography, and networks Sending states and keys, distributing entanglement, and connecting quantum nodes.
Sensing and metrology Estimation, Fisher information, squeezing, interferometry, and noise limits.
Quantum simulation Digital, analog, and hybrid representations of physical models.
Hardware platforms Encodings, native operations, readout, interconnects, and physical constraints.
Software, compilation, and control Translating algorithms into circuits and pulses, and simulating their behavior.
Benchmarking, verification, and validation Learning states and processes, measuring performance, and testing conclusions.
Applications and case studies Comparing complete calculations and experiments with appropriate alternatives.
Frontier connections Stable physical questions behind current research directions.
Reference Gates, states, channels, codes, protocols, metrics, and reading lists.

For the physical mechanisms behind a device, follow the links to AMO physics and Quantum Matter. For state structure and measurement theory, use Composite Systems and Measurement and Open Quantum Systems.

Exercises

Section titled “Exercises”

1. Find the measurement that reaches the discrimination bound

Section titled “1. Find the measurement that reaches the discrimination bound”

For the two states in the worked example, diagonalize ρ0−ρ1\rho_0-\rho_1 and describe which eigenvalue corresponds to each guess.

Solution

In the computational basis,

ρ0−ρ1=12(1−1−1−1)=Z−X2.\rho_0-\rho_1 =\frac12\begin{pmatrix}1&-1\\-1&-1\end{pmatrix} =\frac{Z-X}{2}.

Its eigenvalues are ±1/2\pm1/\sqrt2. Measure the spin component along (−1,0,1)/2(-1,0,1)/\sqrt2, guessing 00 for the positive outcome and 11 for the negative outcome. The trace norm is 2\sqrt2, giving the displayed optimal probability. This measurement exploits a basis other than the sender’s computational basis.

2. Compare coherent and stochastic errors after two uses

Section titled “2. Compare coherent and stochastic errors after two uses”

Starting from ∣0⟩|0\rangle, compare two uses of U=e−iϵX/2U=e^{-i\epsilon X/2} with two independent uses of Ep(ρ)=(1−p)ρ+pXρX\mathcal E_p(\rho)=(1-p)\rho+pX\rho X, where p=sin⁡2(ϵ/2)p=\sin^2(\epsilon/2). Both give the same probability of outcome 11 after one use. Do they agree after two?

Solution

The coherent gates combine to e−iϵXe^{-i\epsilon X}, giving P1=sin⁡2ϵP_1=\sin^2\epsilon. Independent flips give outcome 11 only when exactly one occurs:

P1stoch=2p(1−p)=12sin⁡2ϵ.P_1^{\mathrm{stoch}}=2p(1-p)=\frac12\sin^2\epsilon.

For small ϵ\epsilon, the probabilities are ϵ2\epsilon^2 and ϵ2/2\epsilon^2/2. Agreement in one measurement after one use does not determine how errors compose.

3. Change the allowed circuit connections

Section titled “3. Change the allowed circuit connections”

For the endpoint-line and doubling constructions above, count the CNOTs and entangling layers for sixteen qubits.

Solution

Both use fifteen CNOTs. The specified endpoint-line schedule has fifteen entangling layers; the all-to-all doubling schedule has four. The line count refers to that construction, not to an optimization over the starting site or over measurement-assisted protocols.

References

Section titled “References”