Method, Apparatus and Computer Program Product for Determining the Component of a Magnetic Field in a Predetermined Direction

Abstract

The invention relates to a method for determining the component of a magnetic field in a predetermined direction. The method comprises preparing a quantum system in a coherent superposition state (S1), letting the quantum system evolve for a delay time period (S2) and performing a readout operation and a projective measurement on the quantum system (S3). The steps (S1, S2, S3) are iteratively repeated in an iteration loop, wherein the delay time period increases linearly by the same time increment after each iteration. The method further comprises determining the component of the magnetic field in the predetermined direction according to the outcome of the projective measurements (S4).

Claims

1. Method for determining the component of a magnetic field in a predetermined direction, comprising the steps of: (S1) preparing a quantum system in a coherent superposition state, (S2) letting the quantum system evolve for a delay time period, (S3) performing a readout operation and a projective measurement on the quantum system, and iteratively repeating steps (S1), (S2) and (S3) in an iteration loop IL, wherein the delay time period increases linearly by the same time increment after each iteration, and (S4) determining the component of the magnetic field in the predetermined direction according to the outcomes of the projective measurements.

2. Method according to claim 1, wherein the coherent superposition state is the same in each iteration and/or the coherent superposition state corresponds to an unbalanced superposition of at least three states with respective amplitudes and/or the coherent superposition state corresponds to the maximum modulus spin-projection in a direction perpendicular to the predetermined direction.

3. Method according to claim 1, wherein the readout operation corresponds to a Fourier transform of the state of the quantum system.

4. Method according to claim 1, wherein the projective measurement corresponds to a measurement of the spin polarization of the quantum system in the predetermined direction.

5. Method according to claim 1, wherein the number of iterations is predetermined and/or the number of iterations is determined such that the iteration loop terminates when a delay time period becomes larger than a coherence time of the quantum system and/or the number of iterations is determined such that the total phase accumulation time is larger than at least three times a coherence time of the quantum system.

6. Method according to claim 1, wherein the method further comprises simulating a dynamical evolution of the quantum system using a Hamiltonian and/or a Lindblad master equation and determining the probability distributions of finding the quantum system in a pure state according to the simulation of the dynamical evolution of the quantum system.

7. Method according to claim 1, wherein the method further comprises estimating an initial determination uncertainty for the component of the magnetic field in the predetermined direction and determining the time increment according to the estimate of the initial determination uncertainty and/or estimating an expected information gain and determining the delay time period of the first iteration according to the estimate of the expected information gain.

8. Method according to claim 1, wherein the component of the magnetic field is determined using a Bayesian learning algorithm and/or for each iteration a probability distribution of the component of the magnetic field is updated according to Bayes theorem using a predetermined probability distribution of finding the quantum system in a pure state according to the outcome of the projective measurement and/or the component of the magnetic field is determined from the probability distribution of the component of the magnetic field updated according to Bayes theorem using the result of the projective measurement in the last iteration.

9. Apparatus for determining the component of a magnetic field in a predetermined direction, comprising: a quantum system, and a control and measurement unit, wherein the control and measurement unit is configured to carry out the steps of: (S1) preparing the quantum system (Q) in a coherent superposition state, (S2) letting the quantum system evolve for a delay time period, (S3) performing a readout operation and a projective measurement on the quantum system, and iteratively repeating steps (S1), (S2) and (S3), wherein the delay time period increases linearly by a same time increment after each iteration, and wherein the control and measurement unit is further configured to carry out the step of (S4) determining the component of the magnetic field in the predetermined direction according to the outcomes of the projective measurements.

10. Apparatus according to claim 9, wherein the quantum system corresponds to a d-dimensional qudit with d>2 and/or the quantum system comprises an experimentally controllable energy spectrum, wherein energy level spacings in at least a part of the experimentally controllable energy spectrum depend identically on the component of the magnetic field in the predetermined direction and/or the quantum system is a superconducting circuit.

11. Apparatus according to claim 9, wherein the control and measurement unit further comprises at least one signal generator configured to generate control pulses for preparing the quantum system in the coherent superposition state and/or readout pulses for performing readout operations on the quantum system and/or probe pulses for performing projective measurements on the quantum system.

12. Apparatus according to claim 9, wherein the control and measurement unit further comprises at least one detector and/or is configured to perform quantum non-demolition measurements on the quantum system.

13. Apparatus according to claim 9, wherein the control and measurement unit (CM) further comprises a computing unit and/or an electronic evaluation unit and/or an electronic storage unit.

14. Apparatus according to claim 9, wherein the apparatus further comprises a refrigeration unit configured to cool the quantum system into its ground state.

15. Computer program product comprising a computer program using software means for performing a method according to claim 1 when the computer program runs in a computing unit.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Exemplary embodiments of the invention are illustrated in the drawings and will now be described with reference to FIGS. 1 to 7.

[0044] In the figures:

[0045] FIG. 1 shows a schematic flow diagram of an embodiment of the method,

[0046] FIG. 2 shows a schematic diagram of an embodiment of the apparatus,

[0047] FIG. 3 shows an exemplary pulse sequence corresponding to three iterations,

[0048] FIG. 4 shows an exemplary estimate of the expected information gain,

[0049] FIG. 5 shows exemplary updates of a probability distribution in the iteration loop,

[0050] FIG. 6 shows a comparison of the information gain obtained from different methods,

[0051] FIG. 7 shows the scaling behaviour of different methods.

DETAILED DESCRIPTION OF THE INVENTION

[0052] FIG. 1 shows a schematic flow diagram of an embodiment of the method for determining the component of a magnetic field H.sub.m in a predetermined direction.

[0053] The method comprises a setup step S0. In the setup step S0, a coherent superposition state |ψ.sup.0, a time increment Δt, a delay time period corresponding to a first iteration t.sub.1.sup.L and a number of iterations N are determined. In the setup step S0, a quantum system Q is initially prepared in its ground state.

[0054] The method further comprises preparing the quantum system Q in the coherent superposition state |ψ.sup.0 (preparation step S1), letting the quantum system Q evolve for the delay time period t.sub.i.sup.L (evolution step S2) and performing a readout operation and a projective measurement on the quantum system Q (readout step S3). The preparation step S1, the evolution step S2 and the readout step S3 are iteratively repeated in the iteration loop IL, wherein the delay time period t.sub.i.sup.L increases linearly by the same time increment Δt>0 after each iteration, i.e., t.sub.i.sup.L=t.sub.1.sup.L+(i−1)Δt. The iteration loop IL is parameterized by the iteration loop index i=1, . . . , N. The time period between the preparation step S1 and the readout operation in the readout step S3 is given by the delay time period t.sub.i.sup.L.

[0055] The method further comprises determining the component of the magnetic field in the predetermined direction according to the outcome of the projective measurements (determination step S4).

[0056] FIG. 2 shows a schematic diagram of an embodiment of an apparatus for determining the component of a magnetic field H.sub.m in a predetermined direction. The apparatus comprises a quantum system Q and a control and measurement unit CM. The apparatus further comprises a dilution refrigerator as a refrigerating unit (not shown) configured to cool the quantum system Q into its ground state.

[0057] The control and measurement unit CM is configured to carry out the steps of preparing the quantum system Q in the coherent superposition state |ψ.sup.0 (preparation step S1) and letting the quantum system Q evolve for the delay time period t.sub.i.sup.L (evolution step S2) and performing a readout operation and a projective measurement on the quantum system Q (readout step S3), and iteratively repeating the preparation step S1, the evolution step S2 and the readout step S3, wherein the delay time period t.sub.i.sup.L increases linearly by the same time increment Δt>0 after each iteration, i.e., t.sub.i.sup.L=t.sub.1.sup.L+(i−1)Δt. The control and measurement unit CM is further configured to carry out the step of determining the component of the magnetic field H.sub.m in the predetermined direction according to the outcomes of the projective measurements (determination step S4).

[0058] In an exemplary embodiment, the quantum system Q is a superconducting circuit corresponding to a transmon device. The transmon device comprises a shunting capacitance and a superconducting loop interrupted by two Josephson junctions, wherein the superconducting loop is arranged such that it is threaded by the magnetic flux corresponding to the component of the magnetic field in the predetermined direction. The area of the superconducting loop is predetermined. The transmon device or the source of the magnetic field to be determined are arranged such that the normal vector of the superconducting loop corresponds to the predetermined direction. In the following, the predetermined direction corresponds to the z-direction of a Cartesian coordinate system with its origin corresponding to the geometric center of the superconducting loop. Additionally, the transmon device couples capacitively to a transmission line resonator R.

[0059] The quantum system Q is described by a qutrit (d=3) with three experimentally controllable states |k (k={0, 1, 2}). The qutrit states are the eigenstates of the component of the spin operator in the predetermined direction. Thus, the qutrit states correspond to the three projections −μ, 0, μ of the magnetic moment of the transmon device. Here, the magnetic component μ of the transmon device has been measured in advance and is thus predetermined.

[0060] The Hamiltonian of the quantum system Q is given by Ĥ=Σ.sub.k=0.sup.d−1 (ω)|k(k| with the energies E.sub.k(ω) depending on the reduced magnetic field ω=μH.sub.m/ℏ and Planck's constant ℏ. The functional dependence of the energy level spacings E.sub.k,k+1(ω)=E.sub.k+1(ω)−E.sub.k(ω) on the reduced magnetic field ω has been measured for a reference magnetic field and is predetermined. In particular, the energy level spacings of the quantum system Q are given by E.sub.k,k+1(ω)=√{square root over (8E.sub.J(ω)E.sub.c)}−E.sub.c(k+1) with the Josephson energy E.sub.J(ω) corresponding to the Josephson junctions of the superconducting loop and being sensitive to the component of the magnetic field in the predetermined direction. The charging energy E.sub.c is obtained from the total capacitance of the transmon device. Consequently, the energy level spacings E.sub.k,k+1(ω) depend identically on the reduced magnetic field ω and on the component of the magnetic field in the predetermined direction.

[0061] The control and measurement unit CM comprises a signal generator S. The signal generator S comprises an arbitrary wave generator. The signal generator S is configured to generate two-tone radio-frequency control pulses C.sub.1, C.sub.2, C.sub.3 of rectangular shape for preparing the quantum system Q in the coherent superposition state |ψ.sup.0. The signal generator S is also configured to generate two-tone radio-frequency readout pulses R.sub.1, R.sub.2, R.sub.3 of rectangular shape for performing readout operations on the quantum system Q. Additionally, the signal generator S is configured to generate probe pulses for performing projective measurements on the quantum system Q.

[0062] The control and measurement unit CM comprises a detector D. The control and measurement unit CM and the detector D are configured to perform quantum non-demolition measurements on the quantum system Q. The control and measuring unit CM also comprises coupling means configured to couple the signal generator S with the quantum system Q (e.g., via a gate line) and with the transmission line resonator R (e.g., via a transmission line). The control and measuring unit CM also comprises coupling means configured to couple the detector D with the transmission line resonator (e.g., via a transmission line). Moreover, the control and measurement unit CM comprises an electronic evaluation unit E and an electronic storage unit. The electronic storage unit is part of the electronic evaluation unit E.

[0063] FIG. 3 shows a pulse sequence generated by the signal generator S for the first three iterations (i=1, 2, 3) in the iteration loop IL. The pulse sequence comprises the control pulses C.sub.1, C.sub.2, C.sub.3 for the preparation step S1 and the readout pulses R.sub.1, R.sub.2, R.sub.3 for the readout operation in the readout step S3. The pulse sequence is brought into interaction with the quantum system Q via the coupling means connecting the signal generator S and the quantum system Q. The pulse durations of the control pulses C.sub.1, C.sub.2, C.sub.3 and the readout pulses R.sub.1, R.sub.2, R.sub.3 are considerably smaller than the delay time period t.sub.1.sup.L. The time differences between a control pulse C.sub.1, C.sub.2, C.sub.3 and a readout pulse R.sub.1, R.sub.2, R.sub.3 are given by the delay time periods t.sub.1.sup.L, t.sub.2.sup.L=t.sub.1.sup.L+Δt and t.sub.3.sup.L=t.sub.1.sup.L+2Δt, respectively. In an alternative embodiment, a delay time period t.sub.1.sup.L, t.sub.2.sup.L=t.sub.1.sup.L+Δt and t.sub.3.sup.L=t.sub.1.sup.L+2Δt may also correspond to the time difference between the center of a control pulse C.sub.1, C.sub.2, C.sub.3 and the center of a readout pulse R.sub.1, R.sub.2, R.sub.3.

[0064] For the purpose of performing a quantum non-demolition measurement on the quantum system Q, the quantum system Q is configured to couple to the transmission line resonator R only dispersively. A resonance of the combined system (R and Q) is then probed using a probe pulse (not shown) generated by the signal generator S after a readout pulse R.sub.1, R.sub.2, R.sub.3. The probe pulse is a microwave pulse coupling to the transmission line resonator R via a transmission line. Since the transmission line resonator R is coupled to the quantum system Q, the probe pulse probes the energy spectrum of the combined system (R and Q). The state of the quantum system Q, i.e., the measurement result of the projective measurement, is then obtained from the phase shift of the probe pulse reflected from the transmission line resonator R and measured with the detector D (dispersive readout). After each quantum non-demolition measurement, the quantum system Q relaxes again into its ground state before the next iteration starts.

[0065] In the following, further details and exemplary embodiments are provided:

[0066] In the setup step S0, the amplitudes of the coherent superposition state |ψ.sup.0 are determined such that the coherent superposition state |ψ.sup.0 corresponds to the maximum modulus spin-projection in the x-direction perpendicular to the predetermined z-direction. Specifically, the coherent superposition state is given by |ψ.sup.0 =(|0+√{square root over (2)}|1+|2)/2. Thus, the coherent superposition state |ψ.sup.0 is the eigenstate of the x-component Ĵ.sub.x (or x-projection) of the spin-operator Ĵ with the largest eigenvalue 1 in terms of its absolute value. In the qutrit basis the coherent superposition state |ψ.sup.0 is written as a vector (½, 1/√{square root over (2)}, ½).sup.T. In an alternative embodiment, the coherent superposition state |ψ.sup.0 can be written more generally as e.sup.iβ(e.sup.iλ/2, 1/√{square root over (2)}, e.sup.−1λ/2).sup.T with arbitrary real numbers β, λ.

[0067] In the setup step S0, the initial probability distribution of the component of the magnetic field is determined to be a Gaussian function P.sup.0(ω|∅)=N(ω.sub.R, σ.sup.2) with a mean ω.sub.R and the standard deviation σ. The mean ω.sub.R corresponds to an estimated value for the component of the magnetic field obtained from a classical measurement. The standard deviation σ corresponds to an estimated initial determination uncertainty. The time increment Δt is obtained from the inverse of the estimated initial determination uncertainty σ as Δt=π/σ. In an alternative embodiment, the time increment may also be chosen differently and adapted to the requirements of an apparatus. However, the time increment should ideally not deviate more than an order of magnitude from the value π/σ.

[0068] In the setup step S0, an expected information gain is estimated and the delay time period of the first iteration t.sub.1.sup.L is determined according to the estimate of the expected information gain. More specifically, the delay time period of the first iteration t.sub.1.sup.L is determined to be the saturation time of the expected information gain corresponding to the first iteration. The expected information gain is determined from a simulation of the dynamical evolution of the quantum system Q during the first iteration before the first iteration is actually carried out as explained further below in more detail.

[0069] In general terms, the information gain ΔI.sup.n+1 is defined as a decrease in entropy ΔI.sup.n+1=S.sup.n−S.sup.n+1 with n=0, . . . , N−1. Thereby, the entropy is given by the Shannon entropy in terms of a probability distribution P.sup.n(ω|{ξ.sub.i, t.sub.i, s.sub.i}.sub.i=1.sup.n) of the magnetic field (after n iterations) or, equivalently (up to a trivial variable transformation), the probability distribution of the reduced magnetic field ω, i.e.,

S.sup.n({ξ.sub.i,t.sub.i,s.sub.i}.sub.i=1.sup.n)=−∫P.sup.n(ω′|{ξ.sub.i,t.sub.i,s.sub.i}.sub.i=1.sup.n)ln P.sup.n(ω′|{ξ.sub.i,t.sub.i,s.sub.i}.sub.i=1.sup.n)dω′

[0070] where ξ.sub.i labels a pure state of the quantum system Q corresponding to the outcome of the projective measurement in the i-th iteration, t.sub.i is a time corresponding to a delay time period and s.sub.i denotes a vector of experimental parameters, i.e., the frequencies of the control and readout pulses. Since the preparation step S1 and the readout operation are chosen identically in each iteration, the index on the vector can also be dropped, i.e., s=s.sub.i.

[0071] For n=0, the probability distribution P.sup.n(ω|{ξ.sub.i, t.sub.i, s.sub.i}.sub.i=1.sup.n) corresponds to the initial probability distribution P.sup.0(ω|∅). For n>0, the probability distribution P.sup.n(ω|{ξ.sub.i, t.sub.i, s.sub.i}.sub.i=1.sup.n) is related to the probability distribution P(ξ.sub.n|ω, t.sub.n, s.sub.n) of finding the quantum system Q in the pure state |ξn via Bayes theorem, i.e., via the recurrence

P.sup.n(ω|{ξ.sub.i,t.sub.i,s.sub.i}.sub.i=1.sup.n)=P.sup.n−1(ω|{ξ.sub.i,t.sub.i,s.sub.i}.sub.i=1.sup.n−1)P(ξ.sub.n|ω,t.sub.n,s.sub.n)N.sub.n,

where N.sub.n is a normalization factor.

[0072] In the setup step S0, the probability distributions P({tilde over (ξ)}.sub.1|ω, t.sub.1, s.sub.1) of finding the quantum system Q in the pure state |{tilde over (ξ)}.sub.1 after the readout operation in the first iteration are calculated using a simulation of the dynamical evolution of the quantum system Q as explained further below. The corresponding algebraic expressions for the probability distributions are electronically stored in the electronic storage unit as a function of the reduced magnetic field (treated as a variable) for all possible pure states {tilde over (ξ)}.sub.1={0, 1, 2}.

[0073] More specifically, the dynamical evolution of the quantum system Q is simulated using a Lindblad master equation for the density matrix {circumflex over (ρ)} of the quantum system Q, i.e.,

[00001]$\frac{d\hat{\rho }}{\mathrm{dt}}=-i\left[\hat{\rho },{\hat{H}}_{int}\right]+{\Gamma }_{01}\hat{D}\left[{\sigma }_{01}\right]\hat{\rho }+{\Gamma }_{12}\hat{D}\left[{\sigma }_{12}\right]\hat{\rho }.$

with the superoperator

{circumflex over (D)}[σij]{circumflex over (ρ)}={circumflex over (σ)}ij{circumflex over (ρ)}{circumflex over (σ)}†−½{{circumflex over (σ)}ij†{circumflex over (σ)}ij,{circumflex over (ρ)}}

the Lindblad (jump) operators

{circumflex over (σ)}ij=|ij|(i,j={0,1,2})

and Ĥ.sub.int denoting the Hamiltonian Ĥ written in the quantum-mechanical interaction picture (Dirac picture). In the Lindblad master equation Γ.sub.01 and Γ.sub.12 denote predetermined decoherence rates of the quantum system Q. They are predetermined as Γ.sub.01=Γ.sub.12/{circle around (2)}=Γ, where T.sub.c=1/Γ denotes the predetermined coherence time of the quantum system Q. In an alternative embodiment, one may also include a dephasing term in the Lindblad master equation with a predetermined dephasing rate.

[0074] The probability distributions P({tilde over (ξ)}.sub.1|ω, t.sub.1, s.sub.1) are then obtained from

P({tilde over (ξ)}.sub.1|ω,t.sub.1,s.sub.1)=|{tilde over (ξ)}.sub.1|U.sub.(1).sup.r{circumflex over (ρ)}(t.sub.1)U.sub.(1).sup.r†|{tilde over (ξ)}.sub.1|.sup.2,

[0075] wherein {circumflex over (ρ)}(t.sub.1) is a simulated state of the quantum system Q obtained from a solution of the Lindblad master equation for the initial coherent superposition state {circumflex over (ρ)}.sup.0=|ψ.sup.0ψ.sup.0|. Here, t.sub.1 is treated as a variable for the delay time period. The unitary transformation matrix U.sub.(1).sup.r† corresponds to the readout operation in the first iteration (for an explicit matrix representation see further below).

[0076] In an alternative embodiment, the probability distribution P({tilde over (ξ)}.sub.1|ω, t.sub.1, s.sub.1) are obtained from a simulation using the Hamiltonian for the coherent dynamics of the quantum system Q, i.e.,

P({tilde over (ξ)}.sub.1|ω,t.sub.1,s.sub.1)=|{tilde over (ξ)}.sub.1|ψ.sub.1.sup.f|.sup.2 with |ψ.sub.1.sup.f=.Math..sub.(1).sup.r.Math..sub.(1).sup.e.Math..sub.(1).sup.p|0,

[0077] where the time evolution operator .Math..sub.(1).sup.e=e.sup.−(i/h)Ĥt.sup.1 is obtained from the Hamiltonian Ĥ in the standard way. The unitary transformation matrix U.sub.(1).sup.p† corresponds to (the preparation step S1 in the first iteration (for an explicit matrix representation see further below).

[0078] Since the outcome of a projective measurement is not known before an iteration is actually carried out, the expected information gain ΔI.sup.n+1 is defined in terms of an average over all possible outcomes {tilde over (ξ)}.sub.n+1, i.e.,

ΔI.sup.n+1({ξ.sub.i,t.sub.i,s.sub.i}.sub.i=1.sup.n,{ξ.sub.n+1,t,s})=S.sup.n({ξ.sub.i,t.sub.i,s.sub.i}.sub.i=1.sup.n)−S.sup.n+1({ξ.sub.i,t.sub.i,s.sub.i}.sub.i=1.sup.n,{ξ.sub.n+1,t,s})

where

S.sup.n+1({ξ.sub.i,t.sub.i,s.sub.i}.sub.i=1.sup.n,{ξ.sub.n+1,t,s})=Σ.sub.ξ.sub.n+1.sub.=0.sup.d−1∫S.sup.n+1({ξ.sub.i,t.sub.i,s.sub.i}.sub.i=1.sup.n,{ξ.sub.n+1,t,s})×P(ξ.sub.n+1|ω′,t,s)P.sup.n(ω′|{ξ.sub.i,t.sub.i,s.sub.i}.sub.i=1.sup.n)dω′.

[0079] In the setup step S0, the expected information gain corresponding to the first iteration ΔI.sup.1 is calculated. The integral over the reduced magnetic field variable is estimated using a discretization with respect to the reduced magnetic field corresponding to an evenly spaced grid ω.sub.m with m=1, . . . , M and M=10.sup.5 points according to

[00002]$.Math.\Delta I^1\left({\tilde{\xi }}_1,t,s\right).Math.=-\underset{m=1}{\overset{M}{.Math.}}{P}^0\left({\omega }_m|\varnothing \right)\ln P^0\left({\omega }_m|\varnothing \right)-\underset{{\tilde{\xi }}_1=0}{\overset{2}{.Math.}}\underset{m=1}{\overset{M}{.Math.}}{S}^1\left({\tilde{\xi }}_1,t,s\right)P\left({\tilde{\xi }}_1|{\omega }_m,t,s\right)P^0\left({\omega }_m|\varnothing \right)$$S^1\left({\tilde{\xi }}_1,t,s\right)=-\underset{m=1}{\overset{M}{.Math.}}\frac{P^0\left({\omega }_m|\varnothing \right)P\left({\tilde{\xi }}_1|{\omega }_m,t,s\right)}{{.Math.}_{m^{\prime }=1}^M{P}^0\left({\omega }_{m^{\prime }}|\varnothing \right)P\left({\tilde{\xi }}_1|{\omega }_{m^{\prime }},t,s\right)}\times \ln \frac{P^0\left({\omega }_m|\varnothing \right)P\left({\tilde{\xi }}_1|{\omega }_{m^{\prime }},t,s\right)}{{.Math.}_{m^{\prime }=1}^M{P}^0\left({\omega }_{m^{\prime }}|\varnothing \right)P\left({\tilde{\xi }}_1|{\omega }_{m^{\prime }},t,s\right)}$

[0080] Using these expression, the expected information gain ΔI.sup.1 is calculated for different time arguments t. In the setup step S0, the delay time period t.sub.1.sup.L is determined such that it corresponds to the time at which the expected information gain ΔI.sup.1 starts to saturate, i.e., t.sub.1.sup.L=T.sub.s, where T.sub.s is the saturation time of the expected information gain ΔI.sup.1.

[0081] In an alternative embodiment, and for the quantum system Q corresponding to a qutrit, the delay time period t.sub.1.sup.L can also be obtained from the inverse of the estimated initial determination uncertainty σ as t.sub.1.sup.L=π/(9σ). In another embodiment, and for larger pulse durations of the control and readout pulses, the delay time period t.sub.1.sup.L is determined as t.sub.1.sup.L=max(T.sub.s, T.sub.p), wherein T.sub.p is the pulse duration of the control and readout pulses (assumed to be identical).

[0082] In the preparation step S1, the preparing of the quantum system Q in the coherent superposition state |ψ.sub.0 is achieved using a control pulse C.sub.1, C.sub.2, C.sub.3 interacting with the quantum system Q. In the iteration loop IL, the coherent superposition state |ψ.sub.0 is the same in each iteration. Specifically, the preparation step S1 corresponds to a unitary transformation |ψ.sub.0=.Math..sub.(i).sup.p|0 of the state of the quantum system Q, wherein the unitary transformation matrix is given by

[00003]${\widehat{U}}_{\left(i\right)}^p=\mathrm{Exp}\left(-i\left[\begin{array}{ccc}0& {\Delta }_{1\left(i\right)}^p& 0\\ {\Delta }_{1\left(i\right)}^p& 2{ϵ}_{\left(i\right)}^p& {\Delta }_{2\left(i\right)}^p\\ 0& {\Delta }_{2\left(i\right)}^p& 0\end{array}\right]\right)$

[0083] The matrix elements of .Math..sub.(i).sup.p are obtained from the amplitudes of the coherent superposition state |ψ.sup.0 and correspond to the control pulse frequencies of a control pulse C.sub.1, C.sub.2, C.sub.3. The control pulses C.sub.1, C.sub.2, C.sub.3 feature the same control pulse frequencies in each iteration in the iteration loop IL.

[0084] In the evolution step S2, the quantum system Q evolves dynamically for a time given by the delay time period t.sub.i.sup.L=t.sub.1.sup.L+(i−1)Δt. Thereby, an interaction of the quantum system Q with the component of the magnetic field in the predetermined direction changes the relative phases of the state of the quantum system Q, i.e., |ψ.sup.0.fwdarw.|ψ.sub.i.sup.t, such that the qutrit state |k in the coherent superposition state is expected to transform into e.sup.ikD.sup.0.sup.ωt.sup.i|k during a time t.sub.i=t.sub.i.sup.L, wherein ϕ.sub.k=kD.sub.0ωt.sub.i is the relative phase and D.sub.0=∂E.sub.0,1(ω)/∂ω denotes the derivative of the energy level spacing E.sub.0,1 with respect to the reduced magnetic field ω.

[0085] In the readout step S3, a readout operation is performed on the quantum system Q corresponding to a Fourier transform F.sub.3 of the state of the quantum system Q. In each iteration, the same readout operation is performed. The readout operation corresponds to an interaction of the quantum system Q with a readout pulse R.sub.1, R.sub.2, R.sub.3 before the projective measurement takes place. Specifically, the readout operation corresponds to a Fourier transform of the state of the quantum system Q applied to the qutrit states |n with n={0, 1, 2}, i.e., F.sub.d|n=(1/√{square root over (d)})Σ.sub.k=0.sup.d−1e.sup.−2πink/d|k. Correspondingly, the Fourier transform corresponds to a unitary transformation |ψ.sub.i.sup.f=.Math..sub.(i).sup.r|ψ.sub.i.sup.t of the state of the quantum system Q with the matrix

[00004]${\widehat{U}}_{\left(i\right)}^r=\mathrm{Exp}\left(-i\left[\begin{array}{ccc}0& {\Delta }_{1\left(i\right)}^r& 0\\ {\Delta }_{1\left(i\right)}^r& 2{ϵ}_{\left(i\right)}^r& {\Delta }_{2\left(i\right)}^r\\ 0& {\Delta }_{2\left(i\right)}^r& 0\end{array}\right]\right)$

[0086] The matrix elements of U.sub.(i).sup.r are obtained from the Fourier transform F.sub.3 and correspond to the readout pulse frequencies of a readout pulse R.sub.1, R.sub.2, R.sub.3. The readout pulses R.sub.1, R.sub.2, R.sub.3 feature the same readout pulse frequencies in each iteration in the iteration loop IL.

[0087] In the readout step S3, a projective measurement is performed on the quantum system Q after the readout operation. The projective measurement determines the state of the quantum system |ξ.sub.i. The measurement result ξ.sub.i is stored in the electronic storage unit after each iteration.

[0088] In the determination step S4, the component of the magnetic field H.sub.m is determined using a Bayesian learning algorithm. Specifically, the probability distribution of the component of the magnetic field is updated according to Bayes theorem for each iteration

P.sup.n(ω|{ξ.sub.i,t.sub.i,s.sub.i}.sub.i=1.sup.n)=P.sup.n−1(ω|{ξ.sub.i,t.sub.i,s.sub.i}.sub.i=1.sup.n−1)P(ξ.sub.n|ω,t.sub.n,s.sub.n)N.sub.n.

[0089] The initial probability distribution is determined in the setup step S0. In each update, the probability distribution P(ξ.sub.n|ω, t.sub.n, s.sub.n) is chosen from the probability distributions of finding the quantum system in a pure state calculated in the setup step S0 (and stored in the electronic storage unit) according to the actual outcome ξ.sub.n of the projective measurement. The time argument t.sub.n in each update corresponds to a delay time period t.sub.i.sup.L. The component of the magnetic field H.sub.m is obtained from the mean value of the probability distribution of the component of the magnetic field updated according to Bayes theorem according to the measurement result of the projective measurement obtained in the last iteration.

[0090] FIG. 4 shows an estimate of the expected information gain I.sup.1 for different coherent superposition states and time arguments. Here, ω.sub.R=0, σ=2π/(90 ns) and D.sub.0=1. The saturation time of the expected information gain is T.sub.s=15 ns. Accordingly, the delay time period of the first iteration is t.sub.1.sup.L=15 ns and the time increment is Δt=45 ns. As shown in FIG. 4, the unbalanced coherent superposition state |ψ.sub.0=|ψ.sub.1.sup.0=(|0+√{square root over (2)}|1+|2)/2 leads to an increase of the expected information gain I.sup.1 of about 8 percent as compared to a situation where the quantum system Q is prepared in the balanced coherent superposition state |ψ.sup.0=|ψ.sub.c.sup.0=(|0+|1+|2)/√{square root over (3)}. This increase is attributed to the larger spin-polarization of the quantum system Q in the unbalanced coherent superposition state with ψ.sub.1.sup.0|Ĵ.sub.x|ψ.sub.1.sup.0=1. On the contrary, in the balanced coherent superposition state the expected value of the spin component perpendicular to the predetermined direction is ψ.sub.c.sup.0Ĵ.sub.x|ψ.sub.c.sup.0=2√{square root over (2)}/3≈0.94. An increase of the expected information gain ΔI.sup.1, then corresponds to a decrease of the width of the probability distribution of the component of the magnetic field and thus an increase of the determination accuracy.

[0091] FIG. 5 shows the probability distribution of the magnetic flux in the predetermined direction for the first 6 iterations. The magnetic flux corresponds to the component of the magnetic field in the predetermined direction threading the superconducting loop of the transmon device. The probability distribution of the magnetic flux thus corresponds to the probability distribution of the component of the magnetic field (up to a trivial variable transformation involving the predetermined area of the superconducting loop). In FIG. 5, the magnetic flux quantum is denoted by ϕ.sub.0.

[0092] FIG. 6 shows a simulation of the information gain as a function of the total phase accumulation time t.sub.ϕ=Σ.sub.i=1.sup.Nt.sub.i.sup.L (corresponding to different number of iterations N for fixed delay time periods t.sub.i.sup.L). The results of the proposed method are denoted by LAMA (with linearly increasing delay time periods). The LAMA method is compared with a classical protocol (with constant delay time periods), a Kitaev protocol (with exponentially increasing delay time periods) and a Fourier protocol (with exponentially decreasing delay time periods, wherein the corresponding graph in FIG. 6 starts with the longest delay time period). The maximal number of iterations considered is N=50. In FIG. 6, the actual information gain I is obtained from the width δω(t.sub.ϕ) of the probability distribution of the component of the magnetic field as I≅−ln δω(t.sub.ϕ)+ln δω(0). In FIG. 6, the outcomes of the projective measurement are generated randomly and sampled from a probability distribution. The information gain shown in FIG. 6 is obtained by averaging over the results of 1000 separately simulated iteration loops IL. The results of the proposed method (LAMA) are compared with the other protocols through simulating the operations of the latter in an analogous manner, although the number of iterations has been adapted slightly as required by the underlying algorithms. In FIG. 6, a coherence time T.sub.c=5 μs is predetermined. The results of the Fourier protocol are shown for three different choices of the delay time period in the first iteration. Clearly, the proposed method (LAMA) provides a higher information gain already for total phase accumulation times smaller than the coherence time T.sub.c of the quantum system Q.

[0093] FIG. 7 shows simulation results analogous to FIG. 6, but for different coherence times T.sub.c. The scaling behaviour of the proposed method (LAMA) for T.sub.c=5 μs (filled circles) is compared with the Kitaev protocol for T.sub.c=5 μs (empty circles), T.sub.c=10 μs (empty squares) and T.sub.c=30 μs (empty triangles) on a lin-log plot. The scaling exponent α is obtained from fitting the dependence of the width δω(t.sub.ϕ) of the probability distribution of the component of the magnetic field on the total phase accumulation time t.sub.ϕ, i.e., δω(t.sub.ϕ)=1/(t.sub.ϕ).sup.α. As shown in FIG. 7, the Kitaev protocol approaches the Heisenberg limit for total phase accumulation times t.sub.ϕ smaller than the coherence time T.sub.c, but the scaling exponent almost drops to zero when the total phase accumulation time t.sub.ϕ approaches the coherence time T.sub.c. On the contrary, the proposed method (LAMA) becomes much more efficient when the total phase accumulation time t.sub.ϕ becomes on the order of the coherence time T.sub.c and larger. In fact, a finite coherence time T.sub.c does not pose any notable limitation on the efficiency of the proposed method (LAMA) until the delay time period becomes comparable to the coherence time T.sub.c.

[0094] Features of the different embodiments which are merely disclosed in the exemplary embodiments as a matter of course can be combined with one another and can also be claimed individually.

[0095] All references, including publications, patent applications, and patents, cited herein are hereby incorporated by reference to the same extent as if each reference were individually and specifically indicated to be incorporated by reference and were set forth in its entirety herein.

[0096] The use of the terms “a” and “an” and “the” and “at least one” and similar referents in the context of describing the invention (especially in the context of the following claims) are to be construed to cover both the singular and the plural, unless otherwise indicated herein or clearly contradicted by context. The use of the term “at least one” followed by a list of one or more items (for example, “at least one of A and B”) is to be construed to mean one item selected from the listed items (A or B) or any combination of two or more of the listed items (A and B), unless otherwise indicated herein or clearly contradicted by context. The terms “comprising,” “having,” “including,” and “containing” are to be construed as open-ended terms (i.e., meaning “including, but not limited to,”) unless otherwise noted. Recitation of ranges of values herein are merely intended to serve as a shorthand method of referring individually to each separate value falling within the range, unless otherwise indicated herein, and each separate value is incorporated into the specification as if it were individually recited herein. All methods described herein can be performed in any suitable order unless otherwise indicated herein or otherwise clearly contradicted by context. The use of any and all examples, or exemplary language (e.g., “such as”) provided herein, is intended merely to better illuminate the invention and does not pose a limitation on the scope of the invention unless otherwise claimed. No language in the specification should be construed as indicating any non-claimed element as essential to the practice of the invention.

[0097] Preferred embodiments of this invention are described herein, including the best mode known to the inventors for carrying out the invention. Variations of those preferred embodiments may become apparent to those of ordinary skill in the art upon reading the foregoing description. The inventors expect skilled artisans to employ such variations as appropriate, and the inventors intend for the invention to be practiced otherwise than as specifically described herein. Accordingly, this invention includes all modifications and equivalents of the subject matter recited in the claims appended hereto as permitted by applicable law. Moreover, any combination of the above-described elements in all possible variations thereof is encompassed by the invention unless otherwise indicated herein or otherwise clearly contradicted by context.