Quantum computing

Quantum computing processes quantum information through controlled physical transformations and quantum measurements. In the quantum circuit model, an initial quantum state evolves through quantum gates, and the Born rule predicts the final outcome probabilities. Quantum interference makes the relative quantum phases of amplitudes computationally useful. David Deutsch's 1985 paper developed a model of universal quantum computation.^[ref]

A one-qubit example shows why amplitudes matter. Starting from $|0⟩$, a Hadamard gate gives $(|0⟩+|1⟩)/\sqrt{2}$; a second Hadamard gate returns $|0⟩$. Insert a Pauli Z gate between the two Hadamard gates and the final state is instead $|1⟩$: the intermediate relative sign changes how amplitudes combine. Measuring the intermediate state alone would give equal probabilities in both cases. A useful quantum algorithm arranges such transformations to make information about the required answer accessible through the final quantum measurement.

Liquid-state NMR quantum computing encodes qubits in distinguishable nuclear spins of molecules in solution. Selective radiofrequency pulses implement one-spin control; NMR scalar coupling supplies entangling evolution. Under $H_J=(\pi \hslash J/2)Z_1{Z}_2$, evolution for $1/(2J)$ with $J>0$, followed by $-\pi /2$ rotations about each $z$ axis, produces a Controlled-Z gate up to global phase. Ensemble detection and common pseudo-pure preparations make small demonstrations distinct from scalable pure-state initialization.

A pseudo-pure state is a density matrix of the formwhere $|\psi ⟩$ is a normalized pure state, $I$ is the identity matrix on a $d$-dimensional space and here $0\le p\le 1$. The term $I/d$ is the maximally mixed state. Under a unitary transformation $U$, this term stays fixed and the selected pure state becomes $U|\psi ⟩$. This is why liquid-state NMR quantum computing can follow pure-state-like transformations in part of a highly mixed ensemble.^[ref]

For a traceless matrix representing an observable $O$, direct evaluation of the trace givesThe identity contribution vanishes because $\mathrm{tr}O=0$. Thus $p$ scales the detectable expectation. It is not itself the purity: expanding ${\rho }^2$ and taking its trace yieldsFor example, for two qubits with $p=1/10$, the purity is $103/400$, close to the maximally mixed value $1/4$, although the selected pure state still controls the traceless signal.

The preparation penalty can also be derived explicitly. Take $n$ independent equal spin-one-half systems with one-spin density matrix $(I+\eta Z)/2$, where $Z$ is a Pauli matrix and $0\le \eta \le 1$ is their individual polarization. Their largest population is $a=((1+\eta )/2{)}^n$. A preparation that preserves this population and averages all other populations to equality has $d=2^n$ andIndeed the selected population of $\rho$ is $[1+(d-1)p]/d=a$. In the weak-polarization regime $n\eta \ll 1$, the binomial theorem gives $p\simeq n\eta /(2^n-1)$. If the energy separation is $\hslash \omega$, set $ϵ=\hslash \omega /(k_{B}T)$; a thermal two-level system has $\eta =\tanh (ϵ/2)\simeq ϵ/2$. This gives $p\simeq nϵ/[2(2^n-1)]$, with every parameter defined. Here $\hslash$ is the reduced Planck constant, $k_B$ the Boltzmann constant and $T$ the temperature. The exponential signal loss concerns this specified averaging preparation; it is not a universal bound on every initialization method.

A controlled rotation gate followed by computational-basis dephasing mixes the two target populations in the control-one subspace. If they are $c,d$, their sum is preserved and their difference is multiplied by $\cos \theta$. The operation is not unitary after dephasing. On a two-qubit diagonal state, the population of $|00⟩$ is unchanged when either qubit is the target, because the other control bit is zero.

The requirements are scalable characterized qubits, initialization, long relevant coherence times, a universal quantum gate set, and selective readout. They concern the whole experimental information-processing cycle, including resources needed for quantum error correction.^[ref]

Quantum computation capable of approximating arbitrary finite quantum circuits using a specified universal quantum gate set, with an error tolerance and resource scaling.

The physical Church-Turing principle asserts that a universal physical computing machine can simulate every finitely realizable physical system. It is a claim about physical laws and possible apparatus, stronger in subject matter than a mathematical account of effective symbolic calculation. Its exact force depends on what simulation accuracy and resource use are required.

David Deutsch's 1985 paper formulated a strong version in terms of perfect simulation by finite means and investigated a universal quantum model.^[ref] For example, describing one qubit requires specifying its quantum state and predicting quantum measurement probabilities; copying a short list of observed outcomes does not simulate all its permitted subsequent operations. Approximation to a chosen accuracy and efficient simulation are further, distinct requirements. The principle motivates universal quantum computation without itself being an experimental proof of every proposed universality claim.

A quantum computer is a physical device that performs controlled quantum computation and extracts an output through quantum measurement. A programmable quantum computer combines physical qubits, control hardware and classical processing that specifies operations and interprets repeated measurements.

David P. DiVincenzo organized the central physical requirements into five capabilities: scalable and characterized qubits; initialization in a known quantum state; sufficiently long relevant decoherence times compared with operation times; a universal set of quantum gates; and quantum measurement of selected qubits. These requirements connect an abstract quantum circuit with an experiment: knowing the intended gate sequence is useful only if the apparatus can initialize, control and read the encoded system accurately.^[ref]

The encoding can use different physical systems. Two selected internal levels of a trapped ion, for example, can represent the basis states of a qubit. The remaining levels and environmental couplings then become part of the physical control problem.