Variationally Optimized Measurement Method and Corresponding Clock Based On a Plurality of Controllable Quantum Systems

Abstract

A method of measuring a physical quantity implemented in a hybrid classical-quantum system, the method comprising initializing the plurality of controllable quantum systems in an initial state, applying a set of preparation gates to the plurality of controllable quantum systems for preparing the plurality of controllable quantum systems in a non-classical state, evolving the non-classical state over a time period for obtaining an evolved state of the plurality of controllable quantum systems, applying a set of decoding gates to the plurality of controllable quantum systems in the evolved state, performing a measurement of the plurality of controllable quantum systems, and determining a derived value of the physical quantity based on a mapping function between an outcome of the measurement and the physical quantity on the classical computation system.

Claims

1. A method of measuring a physical quantity, the method being implemented in a hybrid classical-quantum system, the hybrid classical-quantum system comprising a parametrized quantum circuit on the basis of a plurality of controllable quantum systems and further comprising a classical computation system, the method comprising the steps of: a) initializing the plurality of controllable quantum systems in an initial state; b) applying a set of preparation gates to the plurality of controllable quantum systems for preparing the plurality of controllable quantum systems in a non-classical state, c) evolving the non-classical state over a time period for obtaining an evolved state of the plurality of controllable quantum systems; d) applying a set of decoding gates to the plurality of controllable quantum systems in the evolved state; e) performing a measurement of the plurality of controllable quantum systems; and f) determining a derived value of the physical quantity based on a mapping function between an outcome of the measurement and the physical quantity on the classical computation system; wherein the set of preparation gates and the set of decoding gates each comprise non-linear quantum gates suitable for generating a non-classical state of the plurality of controllable quantum systems and each comprise variational quantum gates characterized by variable actions onto controllable quantum systems of the plurality of controllable quantum systems; and wherein the variable actions are variationally optimized to find an extremal value of a cost function, wherein the cost function averages an estimation error of the derived value over a pre-defined expected prior distribution of the physical quantity.

2. The method of claim 1, wherein the cost function is mathematically equivalent to
C=∫dϕϵ(ϕ)P(ϕ) wherein ϕ is the physical quantity, ϵ(ϕ) is the average estimation error for a given value of the physical quantity, and P(ϕ) is the pre-defined expected prior distribution of the physical quantity.

3. The method of claim 2, wherein the estimation error ϵ(ϕ) is the average mean square error of the derived value with respect to an actual value of the physical quantity.

4. The method of claim 3, wherein ϵ(ϕ) is mathematically equivalent to
ϵ(ϕ)=∫dx[(ϕ−ϕ.sub.est(x)].sup.2p(x|ϕ) wherein x is the outcome, ϕ.sub.est(x) is the mapping function mapping the outcome x to the derived value of the physical quantity, ϕ is the actual value of the physical quantity, and p (x|ϕ) is the conditional probability of measuring the outcome x when the actual value of the physical quantity is ϕ.

5. The method of claim 1, wherein the pre-defined expected prior distribution approximates or is mathematically equivalent to a Normal distribution centered around an expected mean value of the physical quantity or the derived value.

6. The method of claim 1, wherein the derived value is a periodic function with respect to changes of the physical quantity, and the pre-defined expected prior distribution is associated with a standard deviation δϕ smaller than a period of the periodic function.

7. The method of claim 1, wherein the derived value is a periodic function with respect to changes of the physical quantity, and the pre-defined expected prior distribution is associated with a standard deviation 67 ϕ greater than 1/N of the period of the periodic function, wherein N is the number of controllable quantum systems.

8. The method of claim 1, wherein the plurality of controllable quantum systems implement a plurality of two-level-systems, and wherein the mapping function maps a difference between the number of controllable quantum systems in an excited state and in a ground state of the plurality of two-level-systems to the derived value of the physical quantity.

9. The method of claim 8, wherein the mapping function is mathematically equivalent to a linear function of the difference at least over a standard deviation of the pre-defined expected prior distribution.

10. The method of claim 1, wherein the physical quantity is an oscillating frequency of electromagnetic radiation interacting with the plurality of controllable quantum systems prior to and after step c), and wherein the derived value is a phase originating from a difference in the oscillating frequency and a resonant frequency of the plurality of controllable quantum systems.

11. A clock comprising: an oscillator for generating electromagnetic radiation associated with an oscillator frequency; a plurality of controllable quantum systems implementing a corresponding plurality of two-level systems, wherein an energy difference of the two-level systems corresponds to a target clock frequency of the clock; a controller configured to: a) initialize the plurality of controllable quantum systems in an initial state; b) apply a set of preparation gates to the plurality of controllable quantum systems, c) permit an evolution of the plurality of controllable quantum systems over a time period; d) apply a set of decoding gates to the plurality of controllable quantum systems; e) determine a measurement outcome of the plurality of controllable quantum systems; and f) determine a feedback onto the oscillator based on a mapping function between the measurement outcome and a derived frequency difference between the oscillator frequency and the target clock frequency associated with the plurality of two-level systems; wherein before and after the evolution of the plurality of controllable quantum systems over the time period, the controller drives a state rotation of each of the plurality of controllable quantum systems using the electromagnetic radiation of the oscillator to implement a Ramsey interferometer; and wherein the set of preparation gates and the set of decoding gates each comprise non-linear quantum gates suitable for generating a non-classical state of the plurality of controllable quantum systems and each comprise variational quantum gates characterized by variable actions onto at least one of the plurality of controllable quantum systems; and wherein values of the variable actions are the result of a variational optimization of the variable actions based on a cost function, wherein the cost function averages an estimation error of the derived value over a pre-defined expected prior distribution of the physical quantity.

12. The clock of claim 11, wherein the plurality of controllable quantum systems are implemented in a corresponding plurality of atoms.

13. The clock of claim 11, wherein the controller drives a global rotation of the states of the plurality of controllable quantum systems by an angle of substantially π/2 to implement the Ramsey interferometer.

14. The clock of claim 11, wherein the pre-defined expected prior distribution corresponds to an expected statistical distribution of the actual value after the evolution over the time period.

15. The clock of claim 11, wherein the set of preparation gates and the set of decoding gates each implement global rotations of the states of the plurality of controllable quantum systems approximating the operator R.sub.μ (θ.sub.1)=exp(—iθ.sub.1J.sub.μ) and a variational non-linear quantum gate selected from the group of a generalized exchange coupling approximating the operator G (t)=exp[−iHt], with H=Σ.sub.k,l=1.sup.Nj.sub.k,lσ.sub.k.sup.μσ.sub.i.sup.ν+Σ.sub.kΔ.sub.kσ.sub.k.sup.ρor H=Σ.sub.k,l=1.sup.Nj.sub.k,lσ.sub.k.sup.μσ.sub.l.sup.μ+Σ.sub.kΔ.sub.kσ.sub.k.sup.νand with j.sub.k,l representing a generalized coupling strength between controllable quantum systems k,l, a one-axis twisting operation of the states of the plurality of controllable quantum systems approximating the operator T.sub.u (θ.sub.2)=exp(−iθJ.sub.u.sup.2), and a Rydberg dressing operation approximating the unitary operator D.sub.u (θ.sub.2)=exp [−iθ.sub.2 (H.sub.u.sup.D/V.sub.0)], with H.sub.u.sup.D being the effective interaction Hamiltonian and V.sub.0 corresponding to the interaction strength, with μ, ν, ρ specifying the axis of rotation about respective variable angles θ.sub.1, θ.sub.2, the variable angles θ.sub.1, θ.sub.2 and j.sub.k,l or a function thereof being the respective variable actions.

16. The clock of claim 11, wherein the set of preparation gates and the set of decoding gates each comprise a number of n.sub.En and n.sub.De layers of quantum gates, respectively, wherein n.sub.En and n.sub.De are positive integer numbers, and wherein each layer comprises at least one non-linear quantum gate and is parametrized by at least one variable action.

17. The clock of claim 16, wherein n.sub.En is equal to or smaller than n.sub.De.

18. A method of optimizing a measurement of a physical quantity with a hybrid classical-quantum system comprising a plurality of controllable quantum systems, the method comprising the steps of: a) initializing a number of variational parameters, the variational parameters parametrizing variable actions of variational quantum gates for acting onto the plurality of controllable quantum systems; b) repeatedly implementing a measurement sequence of known values of the physical quantity using the plurality of controllable quantum systems, the measurement sequence having the steps of: initializing the plurality of controllable quantum systems in an initial state; applying a set of preparation gates to the plurality of controllable quantum systems for preparing the plurality of controllable quantum systems in a non-classical state, evolving the non-classical state for obtaining an evolved state of the plurality of controllable quantum systems evolved according to a select one of the known values; applying a set of decoding gates to the plurality of controllable quantum systems in the evolved state; and determining a measurement outcome of the evolved state for the select one of the known values; wherein the set of preparation gates and the set of decoding gates each comprise non-linear quantum gates suitable for generating a non-classical state of the plurality of controllable quantum systems, and each comprise variational quantum gates characterized by variable actions onto controllable quantum systems of the plurality of controllable quantum systems; c) mapping each of the measurement outcomes to a corresponding derived value of the physical quantity according to a mapping function; d) determining a cost parameter according to a cost function which averages an estimation error between the derived values and the corresponding known values over a pre-defined expected prior distribution of the physical quantity; e) selecting updated variational parameters to reduce the cost parameter; f) iteratively repeating steps b) to e) towards variational parameters associated with a minimized cost parameter.

19. The method of claim 18, wherein selecting updated variational parameters to reduce the cost parameter comprises estimating an energy landscape or a gradient of the cost function with respect to the variational parameters.

20. The method of claim 19, wherein estimating the energy landscape or the gradient comprises repeatedly implementing the sequence b) to e) with shifted variational parameters, the shifted variational parameters comprising a subset of the variational parameters being shifted with respect to a current set of variational parameters.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0098] The following detailed description will best be understood with reference to the drawings, wherein:

[0099] FIG. 1 shows a schematic example of a measurement system for measuring a physical quantity;

[0100] FIG. 2 illustrates an example of a method of measuring a physical quantity;

[0101] FIG. 3 illustrates a flow diagram of a method of variationally optimizing a set of variational parameters according to an example;

[0102] FIGS. 4A, 4B illustrate another example of a measurement system for measuring a physical quantity;

[0103] FIG. 5 illustrates the result of calculations of the performance of a measurement sequence based on a plurality of controllable quantum systems according to an example;

[0104] FIG. 6 shows an example visualization of quantum states for the measurement sequence discussed in connection with the example of FIG. 5; and

[0105] FIG. 7 illustrates an example of a clock based on a plurality of controllable quantum systems.

DETAILED DESCRIPTION

[0106] The disclosure presented in the following written description and the various features and advantageous details thereof, are explained more fully with reference to the non-limiting examples included in the accompanying drawings and as detailed in the description, which follows. Descriptions of well-known components have been omitted so to not unnecessarily obscure the principal features described herein. The examples used in the following description are intended to facilitate an understanding of the ways in which the disclosure can be implemented and practiced. Accordingly, these examples should not be construed as limiting the scope of the claims.

[0107] FIG. 1 shows a schematic example of a measurement system 10 for measuring a physical quantity, the system 10 being illustrated by a flow of a measurement sequence progressing in time from left to right. The system 10 comprises a plurality of controllable quantum systems 12, wherein individual controllable quantum systems 12 are represented by initial quantum states 10>. In the following, reference will be made to two-level systems having basis states 10> and 11>as controllable quantum systems 12, which may be implemented as nuclear or electronic spins, or electronic levels of atoms, although the disclosure is in general not limited to such an implementation.

[0108] The measurement system 10 further implements a parameterized quantum circuit including a set of preparation gates 14 and a set of decoding gates 16 acting on the state of the plurality of controllable quantum systems 12. In addition, the system 10 comprises a detection system 18 for measuring the state of the plurality of controllable quantum systems 12, which may comprise a plurality of detectors for measuring the state of each of the plurality of controllable quantum systems 12, as shown in FIG. 1.

[0109] As illustrated in FIG. 1, the set of preparation gates 14 may prepare the plurality of controllable quantum systems 12 according to a unitary operation U.sub.En and the set of decoding gates 16 may decode the evolved state before the measurement according to a unitary operation U.sub.De. Each of the set of preparation gates 14 and the set of decoding gates 16 may be composed of a plurality of non-linear quantum gates 22a-f and may further comprise rotations 24a-d of the states of individual quantum systems 12 (in following also called single qubit operations).

[0110] The plurality of non-linear quantum gates 22a-f may drive a coherent evolution of the plurality of controllable quantum systems 12, which may be suitable to induce entanglement between at least two and preferably all of the controllable quantum systems 12 (e.g. multi-qubit gates). The plurality of non-linear quantum gates 22a-f are illustrated as quantum gates acting on all states of the controllable quantum systems 12, but may also be composed of a plurality of quantum gates, which may jointly act on all of the controllable quantum systems 12, such as a plurality of finite range interactions, e.g. acting on spatially neighboring controllable quantum systems.

[0111] The rotations 24a-d of the states of the controllable quantum systems 12 may be individual rotations or may be global rotations of the states of the controllable quantum systems 12 by a pre-determined angle.

[0112] The measurement system 10 may be configured to initialize the plurality of controllable quantum systems 12 in an initial state, and may be configured to prepare the plurality of controllable quantum systems 12 in a non-classical state through the action of the set of preparation gates 14 at the beginning of the measurement sequence.

[0113] The measurement system 10 may be configured to evolve the non-classical state according to a Unitary evolution U.sub.ev over a interrogation period 20 for obtaining an evolved state of the plurality of controllable quantum systems 12. During the interrogation period 20, a signal to be measured may be encoded into the non-classical state of the plurality of controllable quantum systems 12. For example, the signal may be encoded by a coherent free evolution of the non-classical state or through the action of an electromagnetic field acting on the plurality of controllable quantum systems 12.

[0114] At the end of the interrogation period 20, the system 10 may be configured to apply the set of decoding gates 16 to the evolved state of the plurality of controllable quantum systems 12, and the measurement system 10 may be configured to record a resulting outcome by measuring the plurality of controllable quantum systems 12. The measurement system 10 may be configured to perform a projective von Neumann measurement of the plurality of controllable quantum systems 12, e.g. onto the basis states |0> and |1> of a two-level system. For example, the outcome may be a sequence of measurement results for the states of the plurality of controllable quantum systems 12, may be a number of controllable quantum systems 12 measured in the excited state, e.g. |1>, or in the ground state, e.g. |0>, or a difference between the number of controllable quantum systems 12 in an excited state and in a ground state.

[0115] A classical computation system (not shown) of the measurement system 10 may receive the measurement outcome and may be configured to compute a derived value of the physical quantity via a mapping function.

[0116] Accordingly, the system 10 may be configured to quantitatively determine the physical quantity based on the Unitary evolution U.sub.ev imparted onto the plurality of controllable quantum systems.

[0117] FIG. 2 illustrates a flowchart of a method of measuring a physical quantity according to an example, which may be implemented in the measurement system 10 of FIG. 1. The method comprises initializing the plurality of controllable quantum systems 12 in an initial state (S10), and applying a set of preparation gates 14 to the plurality of controllable quantum systems 12 for preparing the plurality of controllable quantum systems 12 in a non-classical state (S12). The method further comprises evolving the non-classical state over a time period 20 for obtaining an evolved state of the plurality of controllable quantum systems 12 (S14), and applying a set of decoding gates 16 to the plurality of controllable quantum systems 12 in the evolved state (S16). The method further comprises performing a measurement of the plurality of controllable quantum systems 12 (S18), and determining a derived value of the physical quantity based on a mapping function between an outcome of the measurement and the physical quantity on the classical computation system (S20).

[0118] Preferably, the set of preparation gates 14 and the set of decoding gates 16 are selected to jointly optimize sensitivity and dynamic range of measuring physical quantity. In principle, the set of preparation gates 14 and the set of decoding gates 16 may be tuned for implementing an optimal quantum interferometer (OQI), as theoretically described by Macieszczak et al. (New. J. Phys. 16, 113002). However, in practice it may not be feasible to reproduce a given Unitary operation, such as the operations needed for the OQI, with a restricted set and number of quantum operations and for comparatively large numbers of controllable quantum systems.

[0119] Instead, the inventors propose to construct the set of preparation gates 14 and the set of decoding gates 16 from native resources available in the measurement system 10 for the plurality of controllable quantum systems 12, and to variationally optimize the set of preparation gates 14 and the set of decoding gates 16 on the basis of a suitable cost function.

[0120] For example, the action of quantum gates 22a-24d of the set of preparation gates 14 or the set of decoding gates 16 may be parameterized by variable actions, such as a variable rotation angle of one of the rotations 24a-d of the states of the quantum systems 12. The variable actions may then be optimized in a variational optimization, wherein an optimization strategy may be represented by a cost function attributing a score to the measurement.

[0121] To jointly optimize sensitivity and dynamic range of measuring physical quantity, the inventors propose to optimize the phase estimation accuracy of the measurement based on a corresponding cost function. A finite dynamic range δϕ may be introduced in a Bayesian approach through the stochastic width of a prior distribution P(ϕ) in the cost function.

[0122] The cost function may average an estimation error of the derived value over a pre-defined expected prior distribution of the physical quantity, e.g. according to

C=∫dϕϵ(ϕ)P(ϕ),  (1)

wherein (ϕ) is the physical quantity, ϵ(ϕ) is the average estimation error for a given value of the physical quantity, and P(ϕ) is the pre-defined expected prior distribution of the physical quantity.

[0123] The estimation error ϵ(ϕ) may be the average mean square error of the derived value with respect to an actual value of the physical quantity, e.g. according to

ϵ(ϕ)=∫dx[ϕ−ϕ.sub.est(x)].sup.2p(x|ϕ),  (2)

wherein x is the outcome, ϕ.sub.est(x) is the mapping function mapping the outcome x to the derived value of the physical quantity, ϕ is the actual value of the physical quantity, and p(x|ϕ) is the conditional probability of measuring the outcome x when the actual value of the physical quantity is ϕ.

[0124] The variable actions of the set of preparation gates 14 and the set of decoding gates 16 may then be optimized towards a minimal cost C.

[0125] FIG. 3 illustrates a flowchart of a method for variationally optimizing the variable actions according to an example. The method comprises constructing non-linear preparation and decoding circuits based on available quantum gates (S22), and performing measurements of multiple known values of the physical quantity for a set of variational parameters defining the preparation and decoding circuit (S24). The method then comprises reading out the outcome to determine the cost function (S26), and updating the variational parameters to optimize the cost function (S28).

[0126] The variational parameters parametrize variable actions of variational quantum gates for acting onto the plurality of controllable quantum systems 12. For example, a variational parameter may be a pulse length or a pulse amplitude of a laser pulse for driving a rotation of one of the controllable quantum systems 12, and a corresponding rotation angle may be a variable action corresponding to the variational parameter.

[0127] The method may repeatedly perform steps S24 to S28 until the value of the cost function for the variational parameters is at or close to a global minimum of the cost function. The skilled person will appreciate that the variational parameters may in principle be numerically optimized, e.g. by calculating or simulating the action of the set of preparation gates 14 and the set of decoding gates 16 on the plurality of controllable quantum systems 12. Additionally or alternatively, the variational parameters may be optimized by evaluating the outcome for the variational parameters on a measurement system 10 featuring controllable quantum systems 12.

[0128] Preferably, both the set of preparation gates 14 and the set of decoding gates 16 should have variational degrees of freedom, i.e., comprise variational quantum gates parametrized by variable actions, such that after the variational optimization, both unitary operators U.sub.En, U.sub.De may depend on the prior distribution P(ϕ).

[0129] The available quantum gates may depend on the plurality of controllable quantum systems 12 used to implement the measurement sequence.

[0130] FIG. 4A schematically illustrates an example of a measurement system 10 for measuring a frequency of electromagnetic radiation. The system 10 comprises a plurality of controllable quantum systems 12, whose quantum state may be controllably evolved based on pulses of the electromagnetic radiation, e.g. implemented by trapped atoms or trapped ions. For example, qubits may be implemented in the ions in the ground state |0> and the excited state |1> of a radiative transition of ions trapped in a Paul trap.

[0131] The radiative transition between the ground state |0> and the excited state |1> may have a (resonant) transition frequency at or close to the frequency of the electromagnetic radiation, such that the plurality of controllable quantum systems 12 may act as sensitive probes of the frequency.

[0132] The set of preparation gates 14 and the set of decoding gates 16 implement unitary operators U.sub.En and U.sub.De, and are each composed of a number of n.sub.En and n.sub.De preparation/decoding layers 26, respectively.

[0133] FIG. 4B illustrates an example of the structure of one of the preparation/decoding layers 26, which comprises two non-linear quantum gates 22a, 22b acting on all of the plurality of controllable quantum systems 12, and a global rotation 24a of the states of the plurality of controllable quantum systems 12.

[0134] The illustrated example shows the non-linear quantum gates 22a, 22b as implemented by the one-axis twisting operator T.sub.μ (θ.sub.i)=exp(−iθ.sub.iJ.sub.μ.sup.2), with η=x,y,z specifying the axis of rotation about respective variable twisting angles θ.sub.i the variable twisting angle θ.sub.1 being the ith variable action, and the operator I.sub.μ being given by j.sub.x,y,z=½Σ.sub.k=1.sup.Nσ.sub.k.sup.x,y,z, wherein N is the number of controllable quantum systems and σ.sub.k.sup.x,y,z are the Pauli operators. The one axis twisting operator may be implemented in ion traps via the Mølmer-Sørensen interaction and may induce entanglement between different ions in the ion trap.

[0135] However, the skilled person will appreciate that the one-axis twisting operations may also be exchanged for Rydberg dressing operations approximating the unitary operator D.sub.μ (θ)=exp[−iθ(H.sub.μ.sup.D/V.sub.0)], e.g. as realized in alkyne earth tweezer clocks, with H.sub.μ.sup.D, being the effective interaction Hamiltonian and V.sub.0 corresponding to the interaction strength. Further, the one-axis twisting operations may be exchanged for other non-linear interactions between controllable quantum systems 12, such as exchange couplings between generalized spins or other next-neighbor interactions. Further, for controllable quantum systems 12 implemented in atoms with large spins, the non-linear quantum gate 22a, 22b may prepare individual atoms in non-classical states, e.g. squeezed spin states, and the non-classical state may accordingly not comprise entanglement between different controllable quantum systems 12.

[0136] The global rotation may be a native operation in the plurality of controllable quantum systems 12 or may be implemented by single rotations of all states of the plurality of controllable quantum systems 12, wherein the corresponding quantum operator may be given by R.sub.x(θ.sub.i)=exp(−iθ.sub.iJ.sub.x). Each of the operations, e.g. angles θ.sub.1, may be associated with a variational parameter, such that each layer may feature in principle three variable actions or three degrees of freedom, wherein two of the variational parameters may parametrize the actions of two non-linear entangling gates.

[0137] Although only one exemplary structure of a layer 26 is shown, the skilled person will appreciate that the order of the quantum operations in the layers 26 of the set of preparation gates 14 may be different from, or inverse to, the order of the quantum operations in the layers 26 of the set of decoding gates 16. Moreover, although a specific set of axes is provided in the example, different axes may be selected in some examples.

[0138] Preferably, the arrangement of the quantum gates of the set of preparation gates 14 and the set of decoding gates 16 is selected to result in an antisymmetric phase estimator ϕ.sub.est as a result of the measurement, e.g. substantially fulfilling the relation ϕ.sub.est(ϕ)=−ϕ.sub.est(−ϕ) for values of the phase within the width δϕ of the pre-defined expected prior distribution. For example, the set of preparation gates 14 and the set of decoding gates 16 may be invariant under the spin x-parity transformation, e.g. via imposing P.sub.xU.sub.EnR.sub.y(−π/2)P.sub.x=U.sub.EnR.sub.y(−π/2) and P.sub.xU.sub.DeP.sub.x=U.sub.De, with P.sub.x=R.sub.x(π/2), and the set of preparation gates 14 and the set of decoding gates 16 may be constructed according to

[00003]$\begin{array}{cc}{U}_{\mathrm{En}}=\left[R_x\left({\theta }_{n_{\mathrm{En}}}^{\left(3\right)}\right)T_x\left({\theta }_{n_{\mathrm{En}}}^{\left(2\right)}\right)T_z\left({\theta }_{n_{\mathrm{En}}}^{\left(1\right)}\right).Math.R_x\left({\theta }_1^{\left(3\right)}\right)T_x\left({\theta }_1^{\left(2\right)}\right)T_z\left({\theta }_1^{\left(1\right)}\right)\right]R_y\left(\frac{\pi }{2}\right),& \left(3\right)\end{array}$$\begin{array}{cc}{U}_{\mathrm{De}}=R_x\left(\frac{\pi }{2}\right)\left[T_z\left({\vartheta }_1^{\left(1\right)}\right)T_x\left({\vartheta }_1^{\left(2\right)}\right)R_x\left({\vartheta }_1^{\left(3\right)}\right).Math.T_z\left({\vartheta }_{n_{\mathrm{De}}}^{\left(1\right)}\right)T_x\left({\vartheta }_{n_{\mathrm{De}}}^{\left(2\right)}\right)R_x\left({\vartheta }_{n_{\mathrm{De}}}^{\left(3\right)}\right)\right],& \left(4\right)\end{array}$

wherein θ.sub.i.sup.(j), ϑ.sub.i.sup.(j) are the jth variational parameter of the ith layer of the set of preparation gates 14 and the set of decoding gates 16, respectively. The resulting measurement may comprise optimal sensitivity around phase values of ϕ=0. The skilled person will appreciate that the gate

[00004]$R_x\left(\frac{\pi }{2}\right)$

in the set of decoding gates 16 may be replaced by

[00005]$R_y\left(\frac{\pi }{2}\right),$

e.g. to obtain optimal sensitivity for phase values around

[00006]$\varphi =\pm \frac{\pi }{2}.$

Additionally or alternatively, the set of preparation gates 14, the coherent evolution 20, or the set of decoding gates 16 may comprise a phase shift, e.g. based on the operator R.sub.z(β), to shift the phase value associated with optimal sensitivity by β. The skilled person will further appreciate that other non-linear operators, such as the Rydberg dressing operator D.sub.μ, may satisfy the same symmetry constraints, such that T.sub.x (θ/ϑ), T.sub.Z (θ/ϑ) in Eqs. (3), (4) may be replaced by D.sub.x(θ/ϑ), D.sub.Z (θ/ϑ), respectively, in some examples.

[0139] Following the initialization of the plurality of controllable quantum systems 12, the illustrated measurement sequence comprises an interaction with the electromagnetic interaction and the plurality of controllable quantum systems 12, e.g. a pulse of the electromagnetic radiation. The pulse of the electromagnetic radiation may induce a first global rotation 28 of the states of the plurality of controllable quantum systems 12. The first global rotation 28 may induce superposition states of the ground state |0> and the excited state |1>of the plurality of controllable quantum systems 12, and may lock the phase of the superposition state to the phase of the electromagnetic radiation. For example, the first global rotation may induce a rotation of the states of the plurality of controllable quantum systems 12 by substantially π/2, such as to induce an equal superposition of the ground state |0> and the excited state |1>.

[0140] The superposition states of the plurality of controllable quantum systems 12 may be non-classical, e.g. spin squeezed or entangled by the unitary operator U.sub.En via the further gates of the set of preparation gates 14. The application of the set of preparation gates 14 may be followed by a interrogation period 20 in which the states of the plurality of controllable quantum systems 12 may freely evolve. Over the interrogation period 20, the states of the plurality of controllable quantum systems 12 may accumulate a phase with respect to the phase of the electromagnetic radiation. The value of the phase may be based on the interrogation time 20 and the detuning between the frequency of the electromagnetic radiation and of the transition frequency of the plurality of controllable quantum systems 12, resulting in an evolved state of the plurality of controllable quantum systems 12.

[0141] The evolved state may be subjected to the action of the set of decoding gates 16, which may comprise a second global rotation 30 of the states of the plurality of controllable quantum systems 12. The second global rotation 30 may induce a similar rotation of the states of the plurality of controllable quantum systems 12 as the first global rotation 28, e.g. may correspond to a second π/2 pulse. The set of decoding gates 16 including the second global rotation 30 may project the evolved state onto the ground state |0> or the excited state |1> based on the accumulated phase, such that a proportion of the plurality of controllable quantum systems 12 in either state may be indicative of the accumulated phase.

[0142] For example, the detection system 18 may measure a population of the ground state |0> or the excited state |1> for the plurality of controllable quantum systems 12, e.g. via state selective fluorescence of atoms. For example, the detection system 18 may measure a number N.sub.excited of controllable quantum systems 12 in the excited state |1> or a number N.sub.ground of controllable quantum systems 12 in the ground state |0>.

[0143] Based on the measurement, a classical computation system (not shown) may determine an outcome m, e.g. m=N.sub.excited−N.sub.ground, and may determine an estimated phase ϕ.sub.est via the mapping function ϕ.sub.est(m), which is preferably a strictly monotonic function of the outcome m at least over the stochastic width δϕ of the pre-defined expected prior distribution.

[0144] For example, the mapping function may be a linear function of the outcome, e.g. (p.sub.est=α*m, wherein α may be a pre-defined constant. The constant α may be numerically or empirically optimized for given sets of preparation and decoding gates 14, 16 and for a pre-defined expected prior distribution having a certain stochastic width δϕ. Hence, depending on a stochastic width δϕ of the pre-defined expected prior distribution, a suitable constant α may be selected for optimizing the variable actions as well as for determining the derived value, e.g. the estimated phase ϕ.sub.est.

[0145] In principle, an optimal parameter α may be selected analytically or numerically, e.g. depending on the prior distribution, and the variable actions may be optimized with a fixed parameter α. However, α may equally be analytically or numerically selected for each iteration of the variational optimization, such that the cost function is minimized. Moreover, the mapping function need not be linear, but may also be selected as part of the variational optimization, e.g., the mapping function may be selected as a minimum mean squared error estimator given a realization of the preparation and decoding circuits 14, 16.

[0146] The estimated phase ϕ.sub.est may be a function of the interrogation time 20, which may correspond to a time between applying the first pulse 28 and applying the second pulse 30. Based on the estimated phase ϕ.sub.est and the interrogation time 20 a frequency detuning between the transition frequency and the frequency of the electromagnetic radiation may be determined.

[0147] FIG. 5 illustrates the result of calculations of the performance of a measurement sequence as illustrated in FIG. 4A wherein the variable actions of the parametrized quantum circuit have been optimized according to the cost function in Eqs. (1, 2) with a normal distribution centered on zero phase and having a standard deviation (width) of δϕ≈0.7 as the pre-defined expected prior distribution. The different curves correspond to measurement sequences with a set of preparation gates 14 and a set of decoding gates 16 with different numbers (n.sub.En, n.sub.De) of layers 26 for constructing the unitary operators U.sub.En, U.sub.De, with each layer 26 comprising 3 variational parameters, as given by Eqs. (3), (4).

[0148] The subfigure FIG. 5(a) shows the estimated phase ϕ.sub.est as a function of the actual phase 4), while the subfigure FIG. 5(b) illustrates the phase dependence of the estimation error ϵ(ϕ). The width of the pre-defined expected prior distribution δϕ is indicated by vertical lines. The lines associated with (0, 0) correspond to a measurement without entanglement, i.e. a standard Ramsey interferometer with coherent probe states. The lines associated with “OQI” correspond to the theoretical bound on an optimal quantum interferometer for the pre-defined expected prior distribution, as discussed by Macieszczak et al. As can be seen from the lines associated with the (1, 0), a set of preparation gates 14 can reduce the estimation error ϵ(ϕ) close to zero phase, but the dynamic range of the associated interferometer remains substantially constant. However, as can be seen from the lines associated with the (1, 3), entanglement at the measurement stage as introduced by a set of decoding gates 16 can optimize both the estimation error ϵ(ϕ) and the dynamic range. Further, the measurement sequence may approximate the optimum quantum interferometer at a comparatively low number of coherent operations, e.g. including 12 variational gates, parametrized by 12 variable actions. In some examples, the set of preparation and the set of decoding gates comprise less than 30 variable actions individually or in total.

[0149] FIG. 6 shows a visualization of quantum states for the measurement sequence discussed in connection with the example of FIG. 5. The subfigures (a)-(f) illustrate the quantum states |ψ.sub.ϕ=exp(−iϕJ.sub.z)|ψ.sub.in of the evolved state and quantum measurement operators as Wigner distributions on a generalized Bloch sphere for N=64 and δϕ≈0.7. The first (a,d), second (b,e), and third column (c,f) correspond to (n.sub.En, n.sub.De)=(1, 0), the optimal quantum interferometer, and to a (1, 3) quantum circuit, respectively. Measurement operators are visualized as different contours on the Bloch sphere corresponding to different measurement outcomes. The corresponding optimized states |ψ.sub.ϕ are shown at selected phases ϕ.sub.0=0, ϕ.sub.1=π/3, and ϕ3=2π/3 as filled, dashed, and empty areas, respectively. Subfigure (g) illustrates the measurement probability p(m|ϕ) corresponding to the overlap between the contours of the measurement distribution and the respective state distribution, displayed in the same column.

[0150] The visualization of the quantum state in the first column shows that the example optimized with a (1, 0) parametrized quantum circuit features a spin squeezed state, squeezed along the z-direction. However, the measurement is ambiguous for the phase values of ϕ.sub.1=π/3, and ϕ3=2π/3, as it results in indistinguishable distributions of the measurement probability p(m|ϕ). Conversely, the angles can be resolved in the second and third columns, wherein the shape of the measurement operators shows eigenstates with well defined phases, wherein the contours of the measurement operator overlap favorably with the shape of the non-classical state prepared by the set of preparation gates 14. As can be seen from subfigures FIGS. 3(c), (f), the non-classical state prepared by the set of preparation gates 14 may deviate from a conventional spin-squeezed state, e.g. as prepared by the OQI, but features a twisted shape, and the measurement operator may feature corresponding similarly shaped contours for different outcomes. The twisted shape of the evolved state and the measurement may be a consequence of a restricted gate set available for the variational optimization in the example of FIGS. 4A, 4B.

[0151] Nonetheless, the method illustrated in conjunction with FIGS. 4A, 4B may approximate the OQI even for a low circuit depth, wherein the variational optimization of both the set of preparation gates 14 and the set of decoding gates 16 may compensate for a limited universality of the underlying quantum gates.

[0152] FIG. 7 illustrates a clock 32 based on the measurement sequence illustrated in the example of FIG. 4a, 4B. The clock 32 comprises an oscillator 34 generating electromagnetic radiation 36 at a clock frequency ω.sub.OSC and a plurality of controllable quantum systems 12, implemented as ions in a trap or neutral atoms in tweezers or optical lattices.

[0153] The clock 32 is configured to implement the measurement sequence illustrated in the example of a measurement system 10 in FIG. 4A using pulses 38 of the electromagnetic radiation 36 at least for inducing a first rotation 28 and a second rotation 30 of the states of the plurality of controllable quantum systems 12. The clock 32 is further configured to apply a set of preparation gates 14 and a set of decoding gates 16 to the plurality of controllable quantum systems 12 before and after an interrogation time 20, which may equally be based at least partially on electromagnetic radiation of the oscillator 34.

[0154] The clock further comprises a detection system 20 adapted to measure a state of the plurality of controllable quantum systems 12, e.g. based on state selective fluorescence of the ions, generating an outcome m.

[0155] A classical computation system 40 is configured to map the outcome of the measurement to a phase ϕ through the mapping function ϕ.sub.est(m), and may determine a detuning Δ=ω−ω.sub.OSC between the clock frequency ω.sub.OSC and a resonant transition frequency co of the controllable quantum systems 12. A control signal based on the detuning Δ or based on the phase may be fed back to the oscillator 34 to adjust the clock frequency ω.sub.OSC, thereby stabilizing the oscillator 34 to the transition frequency ω.

[0156] Persons skilled in the art will readily understand that these advantages (as well as the advantages indicated in the summary) and objectives of this system would not be possible without the particular combination of computer hardware and other structural components and mechanisms assembled in this inventive system and described herein. It will be further understood that a variety of programming tools, known to persons skilled in the art, are available for implementing the features and operations described in the foregoing material. Moreover, the particular choice of programming tool(s) may be governed by the specific objectives and constraints placed on the implementation plan selected for realizing the concepts set forth herein and in the appended claims.

[0157] The description in this patent document should not be read as implying that any particular element, step, or function can be an essential or critical element that must be included in the claim scope. Also, none of the claims can be intended to invoke 35 U.S.C. § 112(f) with respect to any of the appended claims or claim elements unless the exact words “means for” or “step for” are explicitly used in the particular claim, followed by a participle phrase identifying a function. Use of terms such as (but not limited to) “mechanism,” “module,” “device,” “unit,” “component,” “element,” “member,” “apparatus,” “machine,” “system,” “processor,” “processing device,” or “controller” within a claim can be understood and intended to refer to structures known to those skilled in the relevant art, as further modified or enhanced by the features of the claims themselves, and can be not intended to invoke 35 U.S.C. § 112(f).

[0158] The disclosure may be embodied in other specific forms without departing from the spirit or essential characteristics thereof. For example, each of the new components described herein, may be modified to suit particular variations or requirements while retaining their basic configurations or relationships with each other or while performing the same or similar functions described herein. The present embodiments are therefore to be considered in all respects as illustrative and not restrictive. Accordingly, the scope of the disclosure can be established by the appended claims rather than by the foregoing description. All changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein. Further, the individual elements of the claims are not well-understood, routine, or conventional. Instead, the claims are directed to the unconventional inventive concept described in the specification.