Non-unitary Operations#
Contrary to unitary gates, non-unitary operations based on measurements make the quantum state collapse. Find in the following sections all the non-unitary operations supported by MIMIQ.
Contents#
Note
As a rule of thumb all non-unitary operations can be added to the circuit using the function push() by giving the index of the targets in the following order:raw:: quantum register index -> classical register index.
Note
Noise can also be interpreted as a non-unitary operations but will not be treated here, check the noise documentation page to learn more about it.
Note
Once a non-unitary operation is added to your circuit the speed of execution might be reduced. This is because in this case the circuit needs to be re-run for every sample since the final quantum state might be different each time. This is always true except for Measure operations placed at the very end of a circuit.
To learn more about this head to the simulation page.
Note
Some features of unitary gates are not available for non-unitary operations, for instance, matrix(), inverse(), power(), control(), parallel().
Measure#
Mathematical definition#
Measurements are defined by a basis of projection operators \(P_k\), one for each different possible outcome \(k\). The probability \(p_k\) of measuring outcome \(k\) is given by the expectation value of \(P_k\), that is
If the outcome \(k\) is observed, the system is left in the state
It is common to measure in the Z basis (\(P_0=\ket{0}\bra{0}\) and \(P_1=\ket{1}\bra{1}\)), but measurements in other bases are possible too.
How to use measurements#
Available measurement operations: Measure, MeasureX, MeasureY, MeasureZ, MeasureXX, MeasureYY, MeasureZZ.
With MIMIQ you can measure the qubits at any point in the circuit (not only at the end of the circuit) using one of the measurement operations (Measure…). You can add it to the circuit like gates using push(), but you will need to precise both the index for the quantum register (qubit to measure) and classical register (where to store the result):
>>> circuit = Circuit()
>>> circuit.push(Measure(), 0, 0)
1-qubit circuit with 1 instructions:
└── M @ q[0], c[0]
This will add a Measure() on the first qubit of the quantum register to the circuit and write the result on the first bit of the classical register. Recall that the targets are always ordered as quantum register -> classical register -> z register. To learn more about registers head to the Registers: quantum/classical/Z-register section.
You can also use iterators to Measure multiple qubits at once, as for gates:
>>> circuit.push(Measure(), range(0, 10), range(0,10))
10-qubit circuit with 11 instructions:
├── M @ q[0], c[0]
├── M @ q[0], c[0]
├── M @ q[1], c[1]
├── M @ q[2], c[2]
├── M @ q[3], c[3]
├── M @ q[4], c[4]
├── M @ q[5], c[5]
├── M @ q[6], c[6]
├── M @ q[7], c[7]
├── M @ q[8], c[8]
└── M @ q[9], c[9]
Note
In the absence of any non-unitary operations in the circuit, MIMIQ will sample (and, therefore, measure) all the qubits at the end of the circuit by default, see simulation page.
Reset#
Available reset operations: Reset, ResetX, ResetY, ResetZ.
A reset operation consists in measuring the qubits in some basis and then applying an operation conditioned on the measurement outcome to leave the qubits in some pre-defined state. For example, Reset() leaves all qubits in \(\ket{0}\) (by measuring in \(Z\) and flipping the state if the outcome is 1).
Here is an example of how to add a reset operation to a circuit:
>>> circuit = Circuit()
>>> circuit.push(Reset(), 0)
1-qubit circuit with 1 instructions:
└── Reset @ q[0]
Importantly, even though a reset operation technically measures the qubits, the information is not stored in the classical register, so we only need to specify the qubit register. If you want to store the result, see the Measure-Reset section.
Note that a reset operation can be technically seen as noise and is described by the same mathematical machinery, see noise page. For this reason, some of the functionality provided by MIMIQ for noise is also available for resets. Here is one example:
>>> Reset().krausoperators()
[P₀(1.0), SigmaMinus(1.0)]
Measure-Reset#
Available measure-reset operations: MeasureReset, MeasureResetX, MeasureResetY, MeasureResetZ.
A measure-reset operation is the same as a reset operation except that we store the result of the measurement, see Measure and Reset sections. Because of that, we need to specify both quantum and classical registers when adding it to a circuit:
>>> circuit = Circuit()
>>> circuit.push(MeasureReset(), 0, 0)
1-qubit circuit with 1 instructions:
└── MR @ q[0], c[0]
Conditional logic#
If statements#
An if statement consists in applying an operation conditional on the value of some classical register. In that sense, it resembles a classical if statement.
In MIMIQ you can implement it using IfStatement, which requires two arguments: an operation to apply and a BitString as the condition (see Bitstrings page for more information):
>>> IfStatement(GateX(), BitString("111"))
IF (c==111) X
Note
At the moment, MIMIQ only allows to pass unitary gates as arguments to an if statement (which makes if statements unitary for now).
To add an IfStatement to a circuit use the push() function. The first (quantum) indices will determine the qubits to apply the gate to, whereas the last (classical) indices will be used to compare against the condition given. For example:
>>> circuit = Circuit()
>>> # Apply a GateX on qubit 1 if the qubits 2 and 4 are in the state 1 and qubit 3 in the state 0.
>>> circuit.push(IfStatement(GateX(), BitString("101")), 0, 1, 2, 3)
1-qubit circuit with 1 instructions:
└── IF (c==101) X @ q[0], c[1,2,3]
Here, an X gate will be applied to qubit 1, if classical registers 2 and 4 are 1, and classical register 3 is 0. Of course, if the gate targets more than 1 qubit, then all qubit indices will be specified before the classical registers, as usual (see circuit page).
Operators#
Mathematical definition#
Operators refer to any linear operation on a state. An operator does not need to be unitary, as is the case of a gate. This means that any \(2^N \times 2^N\) matrix can in principle represent an operator on \(N\) qubits.
Note
Do not confuse operator with operation. In MIMIQ, the word operation is used as the supertype for all transformations of a quantum state (gates, measurements, statistical operations…), whereas an operator is a sort of generalized gate, a linear tranformation.
Operators available in MIMIQ#
Custom operators: Operator
Special operators: DiagonalOp, SigmaPlus, SigmaMinus, Projector0, ProjectorZ0, Projector1, ProjectorZ1, ProjectorX0, ProjectorX1, ProjectorY0, ProjectorY1, Projector00, Projector01, Projector10, Projector11
Methods available: Gate;matrix().
How to use operators#
Operators cannot be applied to a state directly (it cannot be added to a circuit using push()), because that would correspond to an unphysical transformation. However, they can be used within other operations such as ExpectationValue or to create custom noise models with Kraus, see noise and statistical operations pages.
Operators can be used to compute expectation values as follows (see also ExpectationValue):
>>> op = SigmaPlus()
>>> ev = ExpectationValue(op)
>>> circuit = Circuit()
>>> circuit.push(ev, 0, 0)
1-qubit circuit with 1 instructions:
└── ⟨SigmaPlus(1)⟩ @ q[0], z[0]
Similarly, operators can also be used to define non-mixed unitary Kraus channels (see also Kraus).
For example, we can define the amplitude damping channel as follows:
>>> gamma = 0.1
>>> k1 = DiagonalOp(1, math.sqrt(1-gamma)) # Kraus operator 1
>>> k2 = SigmaMinus(math.sqrt(gamma)) # Kraus operator 2
>>> kraus = Kraus([k1,k2])
This is equivalent to
>>> gamma = 0.1
>>> ampdamp = AmplitudeDamping(gamma)
>>> ampdamp.krausoperators()
[D(1, 0.9486832980505138), SigmaMinus(0.31622776601683794)]
Note
Whenever possible, using specialized operators, such as DiagonalOp and SigmaMinus, as opposed to custom operators, such as Operator, is generally better for performance.