Reversing the weak measurement on a qubit
Methods and systems are disclosed for restoring a state of a qubit transformed by a weak measurement to its original state. Unlike traditional methods, in which, the restoration was carried out by way of another weak measurement, the disclosed method uses an additional qubit, referred to as the ancillary qubit, and appropriate Hadamard and CNOT transformation for restoring the original state. Because the disclosed method avoids a second weak measurement, the time for restoration of the original state is considerably reduced.
1. A reversal quantum circuit for restoring a state of a qubit after a weak measurement for a duration of time τ, the qubit having an associated decay rate γ, to its state before the weak measurement, comprising:
an ancillary qubit prepared in state |0 ;
a Hadamard gate comprising an input and an output, wherein the input of the Hadamard gate receives the ancillary qubit; and
a CNOT gate comprising a control input receiving the qubit after the weak measurement, a target input receiving the output of the Hadamard gate, a control output producing a transformed qubit, and a target output producing a transformed ancillary qubit,
wherein a rotation angle associated with the Hadamard gate is a function of the decay rate γ and the duration τ.
2. The reversal quantum circuit of claim 1 , further comprising:
a detector for measuring a state of the transformed ancillary qubit at the target output of the CNOT gate,
wherein the input of the Hadamard gate iteratively receives the ancillary qubit prepared in state |0 and the control input of the CNOT gate iteratively receives the control output instead of the qubit after the weak measurement each time the detector measures state |1 , and
wherein the rotation angle associated with the Hadamard gate is additionally a function of an iteration number n, the iteration number n being equal to a number of times the detector detects state |1 .
3. The reversal quantum circuit of claim 1 , wherein the rotation angle of the Hadamard gate is equal to tau −1 e 2 n−1 γτ .
4. The reversal quantum circuit of claim 1 , wherein the circuit is implemented using a cavity quantum electrodynamics (QED) system.
5. A method for restoring a state of a qubit after a weak measurement of duration τ, the qubit having an associated decay rate γ, comprising:
(a) preparing an ancillary qubit in state |0 ;
(b) applying Hadamard transformation on the ancillary qubit, the Hadamard transformation having a rotation angle associated with it;
(c) applying a CNOT transformation on the qubit after a weak measurement and the Hadamard transformed ancillary qubit, wherein the qubit after a weak measurement is a control qubit and the Hadamard transformed ancillary qubit is a target qubit, to produce a CNOT transformed qubit and a CNOT transformed ancillary qubit;
wherein the rotation angle of the Hadamard transformation is a function of the decay rate γ and the duration τ.
6. The method of claim 5 , further comprising:
(d) measuring a state of the CNOT transformed ancillary qubit;
(e) incrementing by one an iteration number n having an initial value equal to 1;
repeating steps (a)-(e) each time measuring results in a state |1 ,
wherein for each iteration the rotation angle of the Hadamard transformation is also a function of the iteration number n, and
wherein for each iteration, the CNOT transformed qubit serves as a control qubit instead of the qubit after a weak measurement.
7. The method of claim 6 , wherein the rotation angle of the Hadamard transformation is equal to tan −1 e 2 n−1 γτ .