Gates, Circuits, and Computation Models

A quantum circuit represents a controlled information-processing procedure as an ordered composition of preparations, reversible gates, measurements, and classically conditioned operations. A gate is an ideal mathematical map; a laboratory instruction is a noisy physical implementation and is generally described by a quantum channel.

This chapter is the canonical map of the circuit-model language. The pages on single-qubit gates, multi-qubit gates, and universal gate sets own the detailed gate families.

Conventions Before Calculation

This page uses column state vectors acted on from the left. In an expression

$$U_3U_2U_1|\psi\rangle,$$

$U_1$ acts first. For two qubits, the ordered computational basis is

$$|00\rangle, |01\rangle, |10\rangle, |11\rangle,$$

with the left ket label belonging to the first tensor factor. A one-qubit gate on the first factor is $U\otimes I$; on the second it is $I\otimes U$.

These conventions are not universal across software packages. Before moving a matrix, bit string, or circuit between sources, check:

• tensor-factor and bit-string order;
• whether matrices act on column or row vectors;
• whether circuit diagrams are read left-to-right;
• the signs in rotation gates;
• whether equality is exact or only up to global phase.

Common One-Qubit Gates

In the basis $|0\rangle,|1\rangle$, the Pauli gates are

$$X= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \qquad Y= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}, \qquad Z= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}.$$

They satisfy

$$X^2=Y^2=Z^2=I,$$

and, for example, $XY=iZ$. The Hadamard gate is

$$H=\frac{1}{\sqrt2} \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix},$$

with

$$H|0\rangle=|+\rangle, \qquad H|1\rangle=|-\rangle.$$

The phase gates are

$$S= \begin{pmatrix} 1&0\\ 0&i \end{pmatrix}, \qquad T= \begin{pmatrix} 1&0\\ 0&e^{i\pi/4} \end{pmatrix},$$

so $S=T^2$ and $Z=S^2=T^4$.

Rotation Gates and General One-Qubit Unitaries

For $k=x,y,z$,

$$R_k(\theta) =e^{-i\theta\sigma_k/2} =\cos\frac\theta2\,I -i\sin\frac\theta2\,\sigma_k.$$

Thus

$$R_x(\theta)= \begin{pmatrix} \cos(\theta/2)&-i\sin(\theta/2)\\ -i\sin(\theta/2)&\cos(\theta/2) \end{pmatrix},$$

and

$$R_z(\theta)= \begin{pmatrix} e^{-i\theta/2}&0\\ 0&e^{i\theta/2} \end{pmatrix}.$$

Every one-qubit unitary can be decomposed as

$$U=e^{i\alpha}R_z(\beta)R_y(\gamma)R_z(\delta).$$

The leading phase is irrelevant for an isolated unconditional gate on a state ray, but it can become a relative phase when the operation is placed under coherent control.

Controlled Operations

For control qubit $A$ and target system $B$, the controlled-$U$ gate is

$$C(U) =|0\rangle\langle0|_A\otimes I_B +|1\rangle\langle1|_A\otimes U_B.$$

CNOT is the controlled-$X$ gate:

$$\operatorname{CNOT}|a,b\rangle =|a,a\oplus b\rangle.$$

In the declared basis,

$$\operatorname{CNOT} =\begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&0&1\\ 0&0&1&0 \end{pmatrix}.$$

The controlled-$Z$ gate is

$$CZ=\operatorname{diag}(1,1,1,-1),$$

and the two are related by

$$CNOT=(I\otimes H)CZ(I\otimes H).$$

The dedicated semantic audit extends this definition to open and multiple controls, multiplexed branches, branch-relative phase, kickback, and controlled-access assumptions.

Exchange Gates

SWAP exchanges two tensor factors:

$$\operatorname{SWAP}|a,b\rangle=|b,a\rangle,$$

with matrix

$$\operatorname{SWAP} =\begin{pmatrix} 1&0&0&0\\ 0&0&1&0\\ 0&1&0&0\\ 0&0&0&1 \end{pmatrix}.$$

It can be decomposed as

$$\operatorname{SWAP} =\operatorname{CNOT}_{12} \operatorname{CNOT}_{21} \operatorname{CNOT}_{12},$$

where the rightmost CNOT acts first. A physical platform may offer an iSWAP or partial exchange natively, making this textbook decomposition a poor cost model for that hardware.

A Bell-State Preparation

Starting from $|00\rangle$, apply Hadamard to the first qubit and then CNOT:

$$\begin{aligned} |00\rangle &\xrightarrow{H\otimes I} \frac{|00\rangle+|10\rangle}{\sqrt2}\\ &\xrightarrow{\operatorname{CNOT}} \frac{|00\rangle+|11\rangle}{\sqrt2} =|\Phi^+\rangle. \end{aligned}$$

The output is not a product of one-qubit states. Circuit diagrams therefore encode both reversible transformations and the creation of correlations.

Exact, Projective, and Channel Equality

Three notions must be kept separate:

1. Exact matrix equality: $U=V$.
2. Equality up to global phase: $U=e^{i\phi}V$.
3. Equality as an isolated unitary channel: $U\rho U^\dagger=V\rho V^\dagger$ for every $\rho$.

The second implies the third. It does not imply that controlled-$U$ and controlled-$V$ are equivalent, because the phase appears only in the controlled branch and becomes relative. Likewise, equality on a few basis states is not enough to establish operator equality unless linearity and phases have been checked.

From Gate Lists to Computation Models

The standard circuit model specifies:

• input registers and their initialization;
• an allowed set of local and entangling operations;
• gate ordering and possible parallel layers;
• intermediate or final measurements;
• classical records and feedforward;
• an output rule and resource accounting.

Width counts simultaneously represented subsystems. Depth counts sequential layers under a declared parallelism model. Gate count, connectivity, communication, magic-state consumption, and measurement latency can be independent resources.

Alternative models—measurement-based, adiabatic, continuous-variable, bosonic encoded, and topological computation—organize the physical primitive differently. Equivalence in computational power does not imply equal physical overhead or noise sensitivity.

Universality and Compilation

A continuously parameterized set containing arbitrary one-qubit rotations and an entangling two-qubit gate is universal in the usual finite-dimensional circuit sense. Discrete sets such as Clifford+$T$ are approximately universal: arbitrary target unitaries are approximated to a requested error.

The Clifford gates alone are not universal for arbitrary quantum computation, despite their importance in stabilizer codes and fault tolerance. Distinguish:

• an abstract universal gate set;
• a device’s native gates;
• a code’s fault-tolerant logical gates;
• a compiler’s chosen approximation and error metric.

Universal Gate Sets develops exact versus approximate universality and compilation costs.

Ideal Gates and Physical Operations

An ideal gate acts as

$$\rho\longmapsto U\rho U^\dagger.$$

A physical implementation is more generally a channel $\mathcal E$. Its quality cannot be inferred from one matrix alone. Calibration, leakage, crosstalk, drift, state-preparation and measurement errors, and temporal correlations all affect a circuit-level claim.

Postselected and measurement-conditioned branches are trace-nonincreasing maps before normalization. Reset is not a unitary on the reset subsystem. Classical feedforward makes the computation adaptive rather than one fixed unitary product.

Verification Checks

For a proposed gate or identity:

1. Verify $U^\dagger U=I$ for an ideal unitary.
2. Apply the operation to every computational-basis vector.
3. Check one nontrivial superposition to expose relative phases.
4. State tensor-factor and matrix-multiplication order.
5. Decide whether equality is exact, projective, or channel-level.
6. For controlled gates, retain phases that would be global without control.
7. For hardware claims, specify the implemented channel and error metric.

Common Mistakes

• Reading matrix products left-to-right in time.
• Using a two-qubit matrix with an undeclared basis order.
• Calling every relative phase a global phase.
• Treating $Y$ as a classical bit flip and dropping its phases.
• Forgetting the half-angle in $R_k(\theta)$.
• Assuming Clifford gates alone are universal.
• Equating an ideal unitary with a noisy laboratory pulse.
• Treating measurement, reset, or discarded ancillas as a unitary on the visible register.
• Comparing circuit depths without the same connectivity and parallelism assumptions.

Chapter Map

• Reversible Computation owns bijective classical logic, reversible embeddings, garbage and ancilla accounting, uncomputation, and the boundary with unitary quantum gates.
• Circuit Model defines registers, wires, resources, measurements, and classical control.
• Single-Qubit Gates owns common matrices and Bloch-sphere action.
• Multi-Qubit Gates owns entangling and exchange gates.
• Controlled Operations audits projector-controlled blocks, open and multiple controls, multiplexed branches, branch-relative phase, coherent versus classical conditioning, and controlled-access assumptions.
• Measurement in Circuits audits projective measurement nodes, basis rotations, classical record encoding, selected and unread outputs, shot statistics, and postprocessing.
• Mid-Circuit Measurement and Feedforward owns causal measurement-conditioned branching, reset and qubit reuse, deferred-measurement limits, and branch-aware resource accounting.
• Quantum Fourier Transform owns the finite-register Fourier unitary, binary phase factorization, exact and approximate circuits, swap conventions, and the boundary with classical FFT output.
• Phase Kickback owns coherent phase transduction from common eigenstates and character states, clean Boolean and modular conversions, factorization and cleanup checks, and phase-sensitive readout.
• Quantum Oracles owns coherent information-access contracts: domains and promises, full-space actions, supplied inverse, controlled, powered, and family capabilities, licensed equivalences, query accounting, and oracle-specific fair comparison.
• Measurement-Based Quantum Computation owns finite-qubit open-graph patterns, adaptive equatorial measurements, byproduct and Pauli-frame propagation, flow and gflow determinism, logical wire, rotation, and entangling kernels, and MBQC resource accounting.
• Adiabatic Quantum Computation owns finite-dimensional closed-system computational Hamiltonian encodings, accepted ground-subspace decoders, gapped paths and schedules, licensed error certificates, circuit/history-state equivalence, and normalized AQC resource accounting.
• Quantum Annealing owns device-agnostic finite-time driver–problem processes across declared closed, open, thermal, nonadiabatic, paused, reversed, or quenched regimes, including decoded sample distributions, freeze-out hypotheses, embeddings, gauges, and annealing-specific resource accounting.
• Continuous-Variable Quantum Computation owns the abstract mode-computation record: quadrature and energy conventions, Gaussian affine-symplectic circuits and continuous measurement with feedforward, declared non-Gaussian completion, finite-squeezed cluster patterns, approximation metrics, and model-level resource accounting.
• Topological Quantum Computation owns the finite-anyon computation record: declared anyon data and total-charge sector, fusion-space encoding, oriented braid and fusion-measurement program, adaptive frame, induced projective logical operation, leakage, universal completion, verification, and resource accounting.
• Bosonic and Encoded Computation Models owns finite logical programs induced by oscillator controls and instruments: declared encodings, complete physical composition, induced logical channels, leakage and rejection, recovery and frames, truncation, verification, and resource ledgers.
• Quantum Algorithms and Complexity is the next chapter guide: it turns a chosen computation model into a declared problem, access/output/success contract, resource ledger, classical comparator, and evidence claim.
• Universal Gate Sets treats compilation and approximation.
• Quantum Gates is the compact matrix lookup card.

References

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018.
• J. Preskill, Lecture Notes for Physics 219/Computer Science 219: Quantum Computation, California Institute of Technology.
• P. Kaye, R. Laflamme, and M. Mosca, An Introduction to Quantum Computing, Oxford University Press, 2007.
• J. M. Gambetta, J. M. Chow, and M. Steffen, ‘Building Logical Qubits in a Superconducting Quantum Computing System,’ npj Quantum Information 3, 2 (2017).

Exercises

1. A state is acted on first by $H$ and then by $Z$. Which product represents the circuit, and what is the output on $|0\rangle$?

Solution

The later operation appears on the left, so the product is $ZH$. Therefore

$$ZH|0\rangle =Z|+\rangle =|-\rangle.$$

2. Let $V=e^{i\phi}U$. Show that $U$ and $V$ define the same isolated unitary channel but that their controlled versions need not agree up to one global phase.

Solution

For every density operator,

$$V\rho V^\dagger =e^{i\phi}U\rho U^\dagger e^{-i\phi} =U\rho U^\dagger.$$

But

$$C(V)=|0\rangle\langle0|\otimes I +e^{i\phi}|1\rangle\langle1|\otimes U.$$

The phase multiplies only the control-$1$ branch and is relative to the control-$0$ branch. It cannot generally be removed as one global phase.

3. Derive $CNOT=(I\otimes H)CZ(I\otimes H)$.

Solution

Write

$$CZ=|0\rangle\langle0|\otimes I +|1\rangle\langle1|\otimes Z.$$

Conjugating the target by $H$ and using $HZH=X$ gives

$$(I\otimes H)CZ(I\otimes H) =|0\rangle\langle0|\otimes I +|1\rangle\langle1|\otimes X =CNOT.$$

4. Prepare $|\Phi^+\rangle$ with $H$ and CNOT, then compute either reduced density operator.

Solution

The circuit gives

$$|\Phi^+\rangle =\frac{|00\rangle+|11\rangle}{\sqrt2}.$$

Taking either partial trace removes the cross terms and gives

$$\rho_A=\rho_B=\frac{I_2}{2}.$$

The global state is pure and entangled while each qubit separately is maximally mixed.