Quantum Circuits · Quantum PL Wiki

A quantum circuit represents a computation as operations on wires carrying qubits. In the usual drawing convention, time runs from left to right. Gates acting successively compose as linear maps; gates acting on separate systems combine by tensor product. Measurements produce classical outcomes that may control later operations. The underlying mathematical objects are quantum states and operations.

Gates and a Bell-state example

An ideal gate $U$ is unitary: $U^\dagger U=I$ . For a state vector it acts as $\lvert\psi\rangle\mapsto U\lvert\psi\rangle$ ; on a density matrix it acts as $\rho\mapsto U\rho U^\dagger$ . If $U$ executes before $V$ , their combined matrix is $VU$ .

Starting with two qubits in $\lvert00\rangle$ , apply a Hadamard to the first qubit and a controlled-NOT with the first qubit as control:

\lvert00\rangle\xrightarrow{H\otimes I} \frac{\lvert00\rangle+\lvert10\rangle}{\sqrt2} \xrightarrow{\mathrm{CNOT}} \lvert\Phi^+\rangle= \frac{\lvert00\rangle+\lvert11\rangle}{\sqrt2}.

Computational-basis measurements return 00 and 11 with probability $1/2$ each. Each individual qubit has reduced state $I/2$ , while the joint pure state is entangled. The same two measurement probabilities can arise from a classical mixture, so these observations alone do not certify entanglement.

Measurement and classical control

Measurement generally cannot be represented by a unitary on the measured register alone. If projectors $P_m$ describe a measurement and the outcome is ignored, the channel is

\rho\longmapsto\sum_m P_m\rho P_m.

The branches are added as density matrices, not as probability amplitudes. If outcome $m$ instead selects a subsequent unitary $U_m$ , the unconditional output is $\sum_m U_mP_m\rho P_mU_m^\dagger$ . Keeping the classical outcome requires an explicit classical output register.

This distinction matters when translating circuits into quantum programming languages: classical feedback and coherent quantum control have different semantics.

Limitations

Circuit depth, available gates, connectivity, and measurement latency affect compilation and execution. A mathematical circuit specifies ideal behavior; physical gates and readout require an additional error model. Unbounded loops and recursion require richer descriptions than a single finite static circuit.

The Bell-state calculation is an introductory demonstration. It does not establish a hardware implementation's fidelity or prove a larger algorithm. Verification methods provide ways to formulate and check such claims, with explicit assumptions about inputs, observations, and numerical precision.

References

Adriano Barenco et al. Elementary gates for quantum computation. Physical Review A 52, 3457-3467 (1995). DOI: 10.1103/PhysRevA.52.3457.
Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information, 10th Anniversary Edition. Cambridge University Press. DOI: 10.1017/CBO9780511976667.