Gates, Circuits, and Computation Models

A quantum circuit specifies how registers are prepared, transformed and measured to produce an answer. Its wires can become entangled, its measurements can determine later operations, and temporary registers must sometimes be restored before interference will work. This chapter develops that language and compares it with other models of quantum computation.

Helpful background. Qubits and encodings explains the registers being manipulated; unitary operators and operators on composite systems provide the matrix and tensor-product rules.

Enter this chapter

Start with reversible computation to see how ordinary functions become coherent operations, then use the circuit model to combine gates, measurements and classical control. A gate’s matrix describes an ideal operation; a physical implementation generally requires a noisy quantum channel.

Here column state vectors are multiplied from the left, so the rightmost factor in $U_3U_2U_1|\psi\rangle$ acts first. For two qubits, use the ordered basis $|00\rangle,|01\rangle,|10\rangle,|11\rangle$: the left label is the first tensor factor. A gate on that factor is $U\otimes I$. Check these conventions when translating a circuit to software.

A short circuit that creates entanglement

Apply a Hadamard to the first qubit of $|00\rangle$, followed by CNOT with that qubit as control:

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

The first gate creates a superposition; the second correlates its two branches. Each qubit’s reduced state is $I/2$, although the joint state is pure. Measuring both in the computational basis gives $00$ or $11$, each with probability $1/2$. Those probabilities alone would also describe a classical mixture. Applying the inverse circuit distinguishes them: the Bell state returns to $|00\rangle$ with certainty, while the mixture gives $|00\rangle$ or $|10\rangle$ with equal probability.

This example connects three skills used throughout the chapter: multiply gates in time order, retain relative phases, and distinguish a joint state from its measurement statistics.

Choose the next calculation

Question	Pages to read
How do gate matrices act?	Single-qubit gates develops rotations and phase gates; multi-qubit gates treats entangling and exchange operations.
Which operations can be synthesized?	Universal gate sets separates exact synthesis from approximation and explains the role of non-Clifford gates.
How does a quantum control differ from a classical condition?	Controlled operations and phase kickback track coherent branches and relative phase.
How do measurements change a computation?	Measurement in circuits explains outputs and conditional states; mid-circuit measurement and feedforward treats adaptive control and qubit reuse.
How is information supplied and processed coherently?	Quantum oracles defines the available access; quantum Fourier transform turns binary labels into structured phases.
Can measurements drive a computation?	Measurement-based computation uses entangled resource states and adaptive measurement bases.
Can a changing Hamiltonian perform the computation?	Adiabatic computation develops controlled slow evolution; quantum annealing also considers finite-time and open-system behavior.
What changes when the elementary carriers differ?	Continuous-variable computation, bosonic encodings and topological computation explain mode, code-subspace and anyon descriptions.

Universality and physical cost

Arbitrary one-qubit unitaries together with an entangling two-qubit gate can approximate arbitrary qubit unitaries when the required interactions or routing are available. A finite gate alphabet such as Clifford+$T$ is approximately universal: the approximation error matters. Clifford operations alone, with stabilizer inputs and Pauli measurements, do not supply universal quantum computation.

Computational equivalence between models also comes with assumptions about initialization, measurements, precision and overhead. It does not imply identical laboratory cost. Width counts simultaneously used qubits; depth counts sequential gate layers under a stated parallelism and connectivity model. Gate count, measurement latency and error-correction overhead can matter separately. After choosing a model, algorithms and complexity asks what it can accomplish and at what cost.

Exercises

1. Explain why applying $Z$ after $H$ to $|0\rangle$ gives $|-\rangle$, whereas applying them in the opposite order gives $|+\rangle$.

Solution

The first sequence is $ZH|0\rangle=Z|+\rangle=|-\rangle$. The reverse sequence is $HZ|0\rangle=H|0\rangle=|+\rangle$. The operators do not commute. Their outputs have the same computational-basis probabilities but opposite $X$ outcomes.

2. The gates $I$ and $-I$ define the same isolated unitary channel. Compare their controlled versions on $|+\rangle|0\rangle$.

Solution

$C(I)$ leaves the state unchanged. But $C(-I)=Z\otimes I$, so it produces $|-\rangle|0\rangle$. A phase that was global in the unconditional gate becomes relative between control branches and can be measured in the $X$ basis.

References

• J.-L. Brylinski and R. Brylinski, “Universal Quantum Gates”, in Mathematics of Quantum Computation, R. K. Brylinski and G. Chen, eds., Chapman & Hall/CRC, 101–116 (2002).
• P. Kaye, R. Laflamme and M. Mosca, An Introduction to Quantum Computing, Oxford University Press, 2007.
• 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.