Protocols for overlap squared
The OverlapSquared
, or fidelity, of two quantum states \(\ket{\Psi_0}\) and \(\ket{\Psi_1}\) is given by:
Shot-based calculations of the overlap squared are supported by three InQuanto protocols: ComputeUncompute
, SwapTest
, and
DestructiveSwapTest
. In all cases one may calculate the bare fidelity, as above, or with a kernel consisting of a single Pauli string:
where \(h\) is a scalar factor. All of these protocols use a single quantum circuit to perform the measurement.
ComputeUncompute
Let \(\ket{\Psi_0} = U_0 \ket{\bar{0}}\) and \(\ket{\Psi_1} = U_1 \ket{\bar{0}}\), where \(U_0\) and \(U_1\) are state preparation unitaries. The fidelity is hence given by
\(|\langle \bar{0} | U_0^\dagger U_1 | \bar{0} \rangle |^2\). The ComputeUncompute
protocol prepares the state \(U_0^\dagger U_1 \ket{ \bar{0}}\) directly
by concatenating the state preparation unitaries (“computing” the \(\ket{\Psi_1}\) state and “uncomputing” \(\ket{\Psi_0}\)) [27],
generating a circuit of the form:
![../../_images/vacuum.png](../../_images/vacuum.png)
Fig. 11 Measurement circuit generated by ComputeUncompute
.
The full register is measured in the computational basis, and the overlap squared is given by the probability of measuring all qubits in the zero state, \(\ket{\bar{0}}\). See a simple example below:
from sympy import sympify
from inquanto.operators import QubitOperator, QubitOperatorList
from inquanto.states import QubitState
from inquanto.ansatzes import TrotterAnsatz
from inquanto.computables import OverlapSquared
from inquanto.protocols import ComputeUncompute
from pytket.extensions.qiskit import AerBackend
reference = QubitState([1, 1, 0, 0])
theta0, theta1 = sympify("theta0"), sympify("theta1")
ansatz0 = TrotterAnsatz(
QubitOperatorList(QubitOperator("Y0 X1 X2 X3", 1j), theta0),
reference
)
ansatz1 = TrotterAnsatz(
QubitOperatorList(QubitOperator("X0 Z1 Y2 Z3", 1j), theta1),
reference
)
fidelity = OverlapSquared(ansatz0, ansatz1)
protocol = ComputeUncompute(AerBackend(), n_shots=1000)
protocol.build_from({theta0: -0.111, theta1: 2.5}, fidelity)
protocol.run(seed=0)
circuit = protocol.get_circuits()[0]
print("Circuit depth: ", circuit.depth())
print("Num qubits: ", circuit.n_qubits)
fidelity.evaluate(protocol.get_evaluator())
Circuit depth: 19
Num qubits: 4
0.634
SwapTest
The SwapTest
protocol implements the canonical swap test for evaluating the overlap squared [28]. This procedure prepares
both states in parallel registers and performs a SWAP operation between them, controlled on an ancilla qubit in the \(\ket{+}\) state. In general, this circuit takes the form:
![../../_images/csp.png](../../_images/csp.png)
Fig. 12 Measurement circuit generated by SwapTest
.
The probability of measuring the ancilla qubit in the \(\ket{0}\) state is then given by \(p(0) = \frac{1}{2}(1 + |\langle \Psi_0|\Psi_1 \rangle|^2)\), from which the overlap squared is extracted. Similarly to above, this protocol is used as follows:
from inquanto.protocols import SwapTest
protocol = SwapTest(AerBackend(), n_shots=1000)
protocol.build_from({theta0: -0.111, theta1: 2.5}, fidelity)
protocol.run(seed=0)
circuit = protocol.get_circuits()[0]
print("Circuit depth: ", circuit.depth())
print("Num qubits: ", circuit.n_qubits)
fidelity.evaluate(protocol.get_evaluator())
Circuit depth: 45
Num qubits: 9
0.6499999999999999
DestructiveSwapTest
Similarly to the SwapTest
, the DestructiveSwapTest
prepares both states in parallel registers, but does not require an ancilla qubit.
This method performs an effective XOR operation between the states by appending a ladder of CNOT gates between the i-th qubits of each register, and a layer of Hadamard gates on the control
register [29]. In general, this circuit takes the form:
![../../_images/dsp.png](../../_images/dsp.png)
Fig. 13 Measurement circuit generated by DestructiveSwapTest
.
The total phase shift between the states is given as \(\pi \sum_{i=1}^n M_i^1 M_i^2\), where \(M_i^r\) is the measurement outcome on the i-th qubit of the first or second state register. A particular shot is deemed successful if the bitwise AND operation between the two state registers has even parity. The probability of success is given by \(p_s = \frac{1}{2}(1 + |\langle \Psi_0|\Psi_1 \rangle|^2)\), from which the overlap squared is computed. Similarly to above, this protocol is used as follows:
from inquanto.protocols import DestructiveSwapTest
protocol = DestructiveSwapTest(AerBackend(), n_shots=1000)
protocol.build_from({theta0: -0.111, theta1: 2.5}, fidelity)
protocol.run(seed=0)
circuit = protocol.get_circuits()[0]
print("Circuit depth: ", circuit.depth())
print("Num qubits: ", circuit.n_qubits)
fidelity.evaluate(protocol.get_evaluator())
Circuit depth: 12
Num qubits: 8
0.6439999999999999