Estimating a quantum state of a quantum mechanical system
View Patent ↗A method includes performing quantum state tomography from the statistics of a collection of measurements, each of which has only two possible outcomes and has the feature of being a measurement of a single qubit. By carefully choosing the measurements it becomes possible to infer the state of a quantum system from the statistics. Moreover, the function which computes the state from the measurement statistics can be computed efficiently in the dimension of the underlying system. Reconstructing the quantum state is performed in accordance with the following expression: ρ = ( 2 d ∑ i = 1 d 2 - 1 p i P i + ( 1 - p i ) ( 1 - P i ) ) - ( d 2 - 2 d ) I d , where d is the dimension of the quantum mechanical system, ρ is the state of the quantum mechanical system, I d denotes the identity operator, P i is one of the plurality of measurement projectors, and p i is the probability for the measurement projector P i .
1. A method of estimating a quantum state of a quantum mechanical system, the method comprising:
applying a plurality of operations to the quantum mechanical system;
for each of the plurality of operations, taking a single-qubit measurement of the quantum mechanical system, wherein each single-qubit measurement has a binary result;
determining a plurality of measurement projectors corresponding to the plurality of single-qubit measurements;
determining a probability for each of the plurality of measurement projectors; and
reconstructing the quantum state of the system based on the probabilities and the measurement projectors;
wherein reconstructing the quantum state is performed in accordance with the following expression:
ρ
=
(
2
d
∑
i
=
1
d
2
-
1
p
i
P
i
+
(
1
-
p
i
)
(
1
-
P
i
)
)
-
(
d
2
-
2
d
)
I
d
,
where d is the dimension of the quantum mechanical system, ρ is the state of the quantum mechanical system, I d denotes the identity operator, P i is one of the plurality of measurement projectors, and p i is the probability for the measurement projector P i .
2. The method of claim 1 , wherein determining the plurality of measurement projectors includes constructing the plurality of measurement projectors from unitary matrices of order two such that the single-qubit measurements are pair-wise independent.
3. The method of claim 1 , wherein determining the probability for each of the plurality of measurement projectors includes taking the corresponding single-qubit measurement multiple times.
4. The method of claim 1 , wherein the quantum mechanical system includes a plurality of superconducting qubits.
5. The method of claim 1 , wherein the quantum mechanical system includes a plurality of nuclear magnetic resonance (NMR) qubits.
6. The method of claim 1 , wherein the quantum mechanical system is of a dimension that is a power of two, the method comprising:
selecting the plurality of operations applied to the quantum mechanical system by using Clifford operations to transform arbitrary Pauli matrices to a specific Pauli matrix corresponding to each of the plurality of single-qubit measurements in the standard basis.
7. The method of claim 1 , wherein the quantum mechanical system is of an arbitrary dimension, the method comprising:
selecting the plurality of operations applied to the quantum mechanical system from Latin squares and +1/−1 valued Hadamard matrices.
8. The method of claim 1 , wherein each single-qubit measurement is taken in the standard basis by means of a physical apparatus that can address a component of the system, the component having a dimension of two.