Measuring a N-dimensional quantum system

1 . A method for measuring a first N-dimensional quantum system in a given state ρ s using a positive-operator valued measure (POVM) measurement, the POVM measurement being defined by M measurement operators, the method comprising:

writing the M eigenstates of the POVM measurement in a chosen computational basis of the first quantum system, thereby defining a M×N matrix whose rows are determined by the respective POVM eigenstates;

determining a second M-dimensional quantum system into which the first quantum system is embedded;

determining an initial state ρ ini of the second quantum system which is restricted to the given state ρ s in the first quantum system;

extending the M×N matrix by adding M−N columns to the M×N matrix such that the resulting M × M matrix U is unitary;

applying a decomposition scheme on the unitary matrix U to enable an experimental realisation of the unitary matrix U based on the decomposition;

transforming the initial state ρ ini to a state ρ ext of the second quantum system by applying to the initial state ρ ini the M×M matrix U according to the decomposition scheme; and

measuring the second quantum system in its state ρ ext using a projective measurement in a basis of the second quantum system, thereby determining the probability of the occurrence of an outcome m of the M outcomes of the projective measurement as the probability of the occurrence of the outcome m of the M outcomes of the POVM measurement.

2 . The method of claim 1 , wherein:

the decomposition scheme is applied such that the M×M matrix U is implementable by external control pulses, and

the control pulses are implemented by one of microwave and laser pulses.

3 . The method of claim 1 , wherein:

the application of the decomposition scheme is performed such that the M×M matrix U is decomposed into a sequence of rotation matrices, and

the application of their inverse to the M×M unitary matrix U results in a diagonal matrix.

4 . The method of claim 3 , wherein:

the sequence of rotation matrices is provided such that the application of their inverse to the M×M unitary matrix U results in the identity matrix,

the sequence of rotations comprises a first type of rotation and a second type of rotation;

the first type of rotation is a rotation around a given axis with a rotation angle and with a polar angle; and

the second type of rotation is a rotation applying a relative phase.

5 . The method of claim 4 , wherein:

the application of the sequence of rotations is implemented using sequences of one of microwave and laser pulses; and

the first type of rotations is implemented by one of microwave and laser pulses and the second type of rotations is implemented by a phase shift of all subsequent pulses.

6 . The method of claim 1 , wherein the first quantum system is a qubit system and the second quantum system is a qudit system with M=4 eigenstates.

7 . The method of claim 1 , wherein the number of measurement operators M is four.

8 . The method of claim 1 , wherein the POVM measurement is an informationally complete (IC) POVM measurement.

9 . The method of claim 1 , further comprising resetting the second quantum system to the state ρ S after measuring the second quantum system in its state ρ ext .

10 . The method of claim 1 , wherein the states ρ S and ρ ext are one of mixed states and pure states.

11 . The method of claim 1 , wherein adding M−N columns to the M×N matrix includes performing a Gram-Schmidt procedure.

12 . The method of claim 1 , wherein:

the POVM measurement is a hardware implemented POVM measurement; and

the method further comprises correcting a bias in the POVM measurement, including:

estimating, using quantum detector tomography, the M operators implemented by the hardware-implemented POVM measurement; and

correcting, using the estimated hardware-implemented POVM operators, a measured expectation value of an observable.

13 . A quantum system of dimension M comprising:

a transmon device, wherein:

the transmon device is in an initial state ρ ini which is restricted to a state ρ S of a N-dimensional quantum system embedded in the quantum system, where N<M;

the transmon device is configured to receive a sequence of pulses for transforming the initial state ρ ini to a state ρ ext of the quantum system; and

the transmon device is connected to a readout that is configured to perform a projective-valued measure (PVM) of the transmon device in its state ρ ext .

14 . The system of claim 13 , wherein a probability of the occurrence of an outcome m of the M outcomes of the PVM is provided as the probability of the occurrence of the outcome m of the M outcomes of a POVM measurement of the N-dimensional quantum system.

15 . The system of claim 14 , wherein the POVM measurement is an informationally complete (IC) POVM measurement.

16 . The system of claim 13 , wherein the sequence of pulses is a sequence of one of microwave and laser pulses.

17 . The system of claim 13 , wherein the system is configured to be reset to the state ρ S after the PVM of the transmon device in its state ρ ext .

18 . The system of claim 13 , wherein the states ρ S and ρ ext are one of mixed states and pure states.

19 . The system of claim 13 , wherein the dimension of the N-dimensional quantum system N is two and the dimension M is four.

20 . A computer program product comprising a computer readable storage medium having computer readable program code embodied therewith, the computer readable program code executable by a computer to perform operations comprising:

writing the M eigenstates of the POVM measurement in a chosen computational basis of the first quantum system, thereby defining a M×N matrix whose rows are determined by the respective POVM eigenstates;

determining a second M-dimensional quantum system into which the first quantum system is embedded;

determining an initial state ρ ini of the second quantum system which is restricted to the given state ρ S in the first quantum system;

extending the M×N matrix by adding M−N columns to the M×N matrix such that the resulting M × M matrix U is unitary;

applying a decomposition scheme on the unitary matrix U to enable an experimental realisation of the unitary matrix U based on the decomposition;

transforming the initial state ρ ini to a state ρ ext of the second quantum system by applying to the initial state ρ ini the M×M matrix U according to the decomposition scheme; and

measuring the second quantum system in its state ρ ext using a projective measurement in a basis of the second quantum system, thereby determining the probability of the occurrence of an outcome m of the M outcomes of the projective measurement as the probability of the occurrence of the outcome m of the M outcomes of the POVM measurement.