Operations - Gates, Measurements, and More

Operations

Acting on quantum states can be performed either:

• In a "linear algebra" language where unitaries, measurements, and other operations have separate interfaces. This is an explicitly deterministic lower-level interface, which provides a great deal of control over how tableaux are manipulated. See the Stabilizer Tableau Algebra Manual as a primer on these approaches.
• Or in a "circuit" language, where the operators (and measurements and noise) are represented as circuit gates. This is a higher-level interface in which the outcome of an operation can be stochastic. The API for it is centered around the apply! function. Particularly useful for Monte Carlo simulations and Perturbative Expansion Symbolic Results.

In the circuit language, all operations can be applied on a state with the apply! function. Whether they are deterministic and their computational complexity is listed in the table below. A list of lower-level "linear algebra style" functions for more control over how an operation is performed is also given.

Details of Operations Supported by apply!

Unitary Gates

We distinguish between symbolic gates like sCNOT that have specialized (fast) apply! methods (usually just for single and two qubit gates) and general tableau representation of gates like CliffordOperator that can represent any multi-qubit gate.

Predefined unitary gates are available, like sCNOT, sHadamard, etc.

[sCNOT(2,4),sHadamard(2),sCPHASE(1,3),sSWAP(2,4)]

Any arbitrary tableaux can be used as a gate too.

They can be specified by giving a Clifford operator tableaux and the indices on which it acts (particularly useful for gates acting on a small part of a circuit):

SparseGate(tCNOT, [2,4])

The Clifford operator tableaux can be completely arbitrary.

SparseGate(random_clifford(3), [2,4,5])

If the Clifford operator acts on all qubits, we do not need to specify indices, just use the operator.

Noisy Gates

Each gate can be followed by noise applied to the qubits on which it has acted. This is done by wrapping the given gate into a NoisyGate

ε = 0.03 # X/Y/Z error probability
noise = UnbiasedUncorrelatedNoise(ε)
noisy_gate = NoisyGate(SparseGate(tCNOT, [2,4]), noise)

In circuit diagrams the noise is not depicted, but after each application of the gate defined in noisy_gate, a noise operator will also be applied. The example above is of Pauli Depolarization implemented by UnbiasedUncorrelatedNoise.

One can also apply only the noise operator by using NoiseOp which acts only on specified qubits. Or alternatively, one can use NoiseOpAll in order to apply noise to all qubits.

[NoiseOp(noise, [4,5]), NoiseOpAll(noise)]

The machinery behind noise processes and different types of noise is detailed in the section on noise

Coincidence Measurements

Global parity measurements involving single-qubit projections and classical communication are implemented with BellMeasurement. One needs to specify the axes of measurement and the qubits being measured. If the parity is trivial, the circuit continues, if the parity is non-trivial, the circuit ends and reports a detected failure. This operator is frequently used in the simulation of entanglement purification.

BellMeasurement([sMX(1), sMY(3), sMZ(4)])

There is also NoisyBellMeasurement that takes the bit-flip probability of a single-qubit measurement as a third argument.

Stabilizer Measurements

A measurement over one or more qubits can also be performed, e.g., a direct stabilizer measurement on multiple qubits without the use of ancillary qubits. When applied to multiple qubits, this differs from BellMeasurement as it performs a single projection, unlike BellMeasurement which performs a separate projection for every single qubit involved. This measurement is implemented in PauliMeasurement which requires a Pauli operator on which to project and the index of the classical bit in which to store the result. Alternatively, there are sMX, sMZ, sMY if you are measuring a single qubit.

[PauliMeasurement(P"XYZ", 1), sMZ(2, 2)]

Reset Operations

The Reset operations lets you trace out the specified qubits and set their state to a specific tableau.

new_state = random_stabilizer(3)
qubit_indices = [1,2,3]
Reset(new_state, qubit_indices)

It can be done anywhere in a circuit, not just at the beginning.