Quantum-enhanced magnetometry by phase estimation algorithms with a single artificial atom

Phase estimation algorithms are key protocols in quantum information processing. Besides applications in quantum computing, they can also be employed in metrology as they allow for fast extraction of information stored in the quantum state of a system. Here, we implement two suitably modified phase estimation procedures, the Kitaev- and the semiclassical Fourier-transform algorithms, using an artificial atom realized with a superconducting transmon circuit. We demonstrate that both algorithms yield a flux sensitivity exceeding the classical shot-noise limit of the device, allowing one to approach the Heisenberg limit. Our experiment paves the way for the use of superconducting qubits as metrological devices which are potentially able to outperform the best existing flux sensors with a sensitivity enhanced by few orders of magnitude.


Introduction
Phase estimation algorithms are building elements for many important quantum algorithms, 1 such as Shor's factorization algorithm 2, 3 or Lloyd's algorithm 4 for solving systems of linear equations. At the same time, phase estimation is a natural concept in quantum metrology, 5 where one aims at evaluating an unknown parameter λ that typically enters into the Hamiltonian of a probe quantum system and defines its energy states E n (λ). In a standard (classical) measurement, the precision δλ is restricted by the shot-noise limit δλ ∝ where t is the measurement time. This, however, is not a fundamental limit: in principle, the ultimate attainable precision scales as δλ ∝ 1/t, constrained only by the Heisenberg relation ∆E(λ) ≥ 2π /t, where ∆E = max n,m (E n − E m ). The Heisenberg limit can be achieved with the help of entanglement resources, e.g., using NOON photon states in optics. [6][7][8] However, these states are difficult to create in general and they typically have a short coherence time. Alternatively, one can reach the Heisenberg limit without exploiting entanglement, by using the coherence of the wavefunction of a single quantum system as a dynamical resource.
However, the uncontrollable interaction of the probe with the environment limits the time scale t where the Heisenberg scaling can be attained by the probe's coherence time t ∼ T 2 . A further improvement then has to make use of an alternative measurement strategy with a precision following the standard quantum limit but with a better prefactor.
The unknown parameter λ can be estimated from the phase φ = ∆E(λ) τ / accumulated by the system in the course of its evolution during the time τ ∼ T 2 . The 2π-periodicity of the phase limits the probe's measurement range ∆λ where λ can be unambiguously resolved within the narrow interval [δλ] H = 2π /(µT 2 ), with µ ≡ ∂∆E/∂λ denoting the sensitivity of the probe's spectrum. Therefore, the improvement in the precision at larger T 2 is concomitant with a proportional reduction of the measurement range ∆λ. The use of phase estimation algorithms then allows to resolve the 2π phase uncertainty and hence break this unfavorable trade-off between the measurement precision δλ and the measurement range ∆λ. Moreover, a metrological procedure based on a phase estimation algorithm is Heisenberg-limited: it attains the resolution δλ ∼ [δλ] H within a large measurement range ∆λ [δλ] H with a Heisenberg scaling in the phase accumulation time τ , i.e., δλ ∝ /(µτ ) for τ ≤ T 2 . At larger times τ > T 2 , the measurement proceeds with independent measure-ments involving the optimal time delay τ = T 2 . Running N = t/T 2 experiments and averaging over N 1 outcomes, one can further improve the precision within the standard quantum limit, 9 δλ ∝ 2π /(µT 2 √ N ) ≡ 2π /(µ √ tT 2 ).
There are two major classes of phase estimation algorithms, one suggested early on by Kitaev 10 and a second originating from the quantum Fourier transform. 11,12 In quantum computing, the Kitaev algorithm was run as part of Shor's factorization algorithm 13 and the Fourier transform algorithm was used in optics to measure frequencies. 14 These algorithms are system-independent and can be employed in a variety of experimental settings, e.g., using NV centers in diamond for the sensitive detection of magnetic fields. [15][16][17] Results Here, we implement a modified version of these algorithms using an artificial atom or qubit in the form of a superconducting transmon circuit. 18 We show that the transmon can be operated as a dc flux magnetometer with Heisenberg-limited sensitivity. The sensitivity is boosted by a magnetic moment that is about five orders of magnitude larger than that of natural atoms or ions. The idea of the experiment is to combine the extreme magnetic-field sensitivity of superconducting quantum interference devices (SQUIDs) with an enhanced performance brought about by exploiting quantum coherence. The 'quantum' in the name of this device refers to the macroscopic complex wave function of the superconducting electronic state. In the SQUID loop geometry, the relative phase of the superconducting wavefunctions across the Josephson junctions acquires a dependence on magnetic flux Φ via the Aharonov-Bohm effect. However, despite its quantum origin, in standard SQUID measurements this phase is a classical variable. In contrast, for the SQUID loop of a transmon qubit, the phase turns into a fully dynamical quantum observable and the flux Φ dependence is encoded in the energylevel separation ω 01 (Φ) between the ground state and the first excited state. Therefore, it is possible to exploit the phase difference φ = [ω d − ω 01 (Φ)]τ acquired during a time τ by the qubit when it is prepared into a coherent superposition of the ground and excited energy states and driven by an external microwave field at a frequency ω d . Differently from their "natural" counterparts, where the characteristics of the quantum sensor are sample independent and defined by the atomic structure, for artificial-atom systems, such as the transmon, we need to adapt the algorithms by including device-specific properties in a so-called passport -a sample specific Ramsey interference pattern obtained in advance from characterization measurements, see Fig. 1b.
Making use of phase estimation algorithms, we demonstrate an enhanced dc-flux sensitivity of the transmon sensor in an enlarged flux range as compared to standard (classical) measurement schemes. Recently, a standard measurement procedure using a flux qubit has been used for the measurement of an ac-magnetic field signal. 19 The experiment employs a superconducting circuit in a transmon configuration, consisting of a capacitivelyshunted split Cooper-pair box coupled to a λ/4-wavelength coplanar waveguide (CPW) resonator realized in a 90 nm thick aluminum film deposited on the surface of a silicon substrate, see Fig. 1 and SI 1 for an image of the sensor device. The SQUID loop of the transmon has an area of S 600 µm 2 , which is chosen large in comparison with standard transmon qubit designs in order to provide a higher sensitivity to magnetic-field changes. The magnetic moment of this artificial atom is µ = S |dω 01 /dΦ|, directly proportional to the area The qubit is coupled via a gate capacitor C g to a coplanar waveguide resonator (CPW, in green) with a resonance frequency ω r around 2π × 5.12 GHz. The magnetic flux Φ through the transmon's SQUID loop is controlled by a dc-current flowing through a flux-bias line (in red). An arbitrary waveform generator (AWG) and a microwave analog signal generator are employed to create a Ramsey sequence of two π/2 microwave pulses at a carrier frequency ω d = 2π × 7.246 GHz separated by a time delay τ . The sequence drives the transmon into a superposition of ground and excited states where the state amplitudes depend on the accumulated phase φ = [ω d − ω 01 (Φ)]τ . The qubit state is read out nondestructively using a probe pulse sent to the CPW resonator; the reflected signal is downconverted (not shown in the figure), digitized, and analyzed by a computer. Next, the computer updates a flux distribution function P(Φ) stored in its memory, determines the next optimal Ramsey delay time, and feeds it back into the AWG. a) Qubit transition frequency ω 01 (Φ) as a function of magnetic flux Φ (parabolic curve). The bottom inset shows the CPW resonator's spectrum. The red circles indicate the bias point of our transmon sensor: we operate far away from the 'sweet spot' in a regime where the transmon's frequency ω 01 (Φ) is an approximately linear function of the flux Φ within the entire flux range ∆Φ. For the fluxes around the point considered here, the frequency ω r of the readout CPW resonator remains approximately constant. b) A premeasured sample-specific Ramsey interference fringes pattern defines the 'passport' function of our sensor. This can be regarded as a non-normalized probability function P p (τ, Φ) to observe the qubit in the excited state after a Ramsey sequence with a delay τ for a specific value of the magnetic flux Φ. The largest flux value used to obtain the Ramsey interference fringes pattern Φ = 0.1394 Φ 0 corresponds to a frequency detuning ∆ω = ω d − ω 01 (Φ) = 2π × 15.8 MHz between the drive and the qubit transition frequencies. The flux range of the 'passport' ∆Φ ∼ 2.5 × 10 −3 Φ 0 corresponds to a range 2π × 13.8 MHz in frequency detuning.
S and the rate of change with flux Φ of the transition frequency ω 01 . For our device, we obtain dω 01 /dΦ = − 2π × 5.3 GHz/Φ 0 at the bias point, resulting in µ = 1.10 × 10 5 µ B , where µ B is the Bohr magneton. By comparison, the Zeeman splitting due to the magnetic moment of NV centers is 28 GHz T −1 , corresponding to a magnetic moment of 2µ B . The sample is thermally anchored to the mixing chamber plate of a dilution refrigerator and cooled down to a temperature of roughly ∼ 20 mK. The qubit has a separate flux-bias line and a microwave gate line, the former allowing to change the qubit transition frequency, while the latter is used for the qubit's state manipulation. The qubit state is determined by a nondemolition read-out technique (see Methods and SI 1) measuring the probe signal reflected back from the dispersively-coupled CPW resonator. To increase the magnetic field sensitivity, we bias the qubit away from the 'sweet spot', see Fig. 1a.
This follows an opposite strategy as compared to the situation where the phase estimation algorithms are employed for quantum computing and simulations; in the latter cases, the qubit sensitivity to flux noise is maximally suppressed by tuning the device to the 'sweet spot' characterized by a vanishing first derivative of the energy with respect to flux. Operating away from the 'sweet spot' leads to a reduction of the T 2 time. The is the sum of the relaxation (2T 1 ) −1 and dephasing T −1 φ rates. 20 The dephasing rate appreciably increases at our bias point, which reduces T 2 and thus the number of available steps that can be implemented in the Kitaev-and Fourier algorithms.
In the experiment, we apply a Ramsey sequence of two consecutive π/2 pulses separated by a time delay τ , which corresponds to an effective spin-1/2 precession around the z-axis of the Bloch sphere. The precession angle φ = ∆ω(Φ)τ is defined by the frequency mismatch ∆ω(Φ) = ω d − ω 01 (Φ) between the transition frequency ω 01 (Φ) of the transmon qubit and the fixed drive frequency ω d of the π/2 pulses. The Ramsey sequence drives the transmon from its ground state into a coherent superposition of ground-and excited states with relative amplitudes determined by the phase φ. The theoretical probability to find the transmon in the first excited state is given by and depends both on the delay time τ and on the magnetic flux Φ through the frequency mismatch ∆ω(Φ).
The decay function γ(τ ) accounts for qubit dephasing, typically due to charge or flux noise. By design, the transmon artificial atom is rather insensitive to background charge fluctuations. On the other hand, intrinsic 1/f magnetic-flux noise couples to the SQUID loop and is known to be a relevant source for dephasing in flux qubits [21][22][23] ; in addition, other decoherence mechanisms can be present, see below for details. The dephasing process can be described through an external classical noise source, see Methods. The particular shape of γ(τ ) depends on the noise spectral density at low frequencies. 'White' noise with a constant power density results in an exponential decay function γ(τ ) = exp(−Γ wn τ ), while 1/f -noise produces a Gaussian decay γ(τ ) = exp −(Γ 1/f τ ) 2 . We fit our experimental curves P (τ, ∆ω(Φ)) by Eq. (1) using both an exponential and a Gaussian decay, see SI 2. For our sample with a relaxation time T 1 of about 260 ns, we cannot distinguish between these two fits, neither in the 'sweet spot' nor in the bias point. Fitting the Ramsey oscil-lation at different fluxes one finds Γ −1 wn ≈ 1250 ns and Γ −1 1/f ≈ 780 ns at the 'sweet spot'. At the bias point, these pure dephasing times reduce to 520 ns and 420 ns, respectively. The decay rates Γ wn and Γ 1/f in the bias point then can be translated into equivalent white and 1/f flux noises and we find the spectral densities The function γ(τ ) determines the optimal delay time τ where the sensitivity of the probability P (τ, ∆ω) to the changes in ∆ω and hence to a flux is the highest. In the standard (classical) measurement approach, a minimal delay τ = τ 0 T 2 sets the frequency range ∆ω(Φ) ∈ [0, π/τ 0 ] where the phase φ and hence P (τ, ∆ω) can be unambiguously resolved. This defines the range ∆Φ = π(τ 0 dω 01 /dΦ) −1 where the magnetic flux can be resolved with a precision scaling given by the standard quantum limit (see Methods), where t is the total measurement or sensing time of the experiment and T rep is the time duration of a single Ramsey measurement. A better flux sensitivity can be attained at larger delays τ , where the probability P [τ, ∆ω(Φ)] is more sensitive to changes in ∆ω. We obtain the best sensitivity at τ = τ * defined by the con- The amplitudes A class and A quant in Eqs. (2) and (3) quantify the magnetic flux sensitivities. Measuring at the optimal delay τ = τ * improves the flux resolution by a factor A class /A quant = τ * /(eτ 0 ), which depends on the qubit's coherence time, the latter serving as the quantum resource in our algorithms. Another important factor which enhances the flux sensitivity is the slope dω 01 /dΦ of the transmon's spectrum. At our working point ω 01 = 2π × 7.246 GHz, we have dω 01 /dΦ = − 2π × 5.3 GHz/Φ 0 . The minimal delay is given by τ 0 ≈ 31.6 ns, see SI 1. The repetition time T rep = 6.546 µs involves the maximal time duration of the Ramsey sequence, the duration of the probe pulse (2 µs) and the transmon's relaxation time back into its ground state (4 µs, which is 15 times longer than the T 1 time). Combining these numbers and setting τ * ∼ 2T 1 , we estimate the theoretical value of flux sensitivity for our transmon sensor as A quant 4 × 10 −7 Φ 0 Hz −1/2 , see Eq. (3), providing an improvement by a factor A class /A quant ∼ 6 over the classical sensitivity. Note, that the best sensitivity is attained at T rep = τ * (i.e., for a very fast control and readout) that gives for our sample Measuring at large time delays τ ∼ T 2 leaves an uncertainty in ∆ω(Φ) due to the multiple 2π-winding of the accumulated phase, thereby squeezing the flux range ∆Φ ∼ 2.5 × 10 −3 Φ 0 by the small factor τ 0 /T 2 . The Kitaev-and Fourier phase estimation algorithms, avoid this phase uncertainty by measuring the probability P (τ, ∆ω) at different delays τ k = 2 k τ 0 for K ∼ log 2 (T 2 /τ 0 ) consecutive steps k = 0, . . . , K − 1. As a result, such a metrological procedure is able to resolve the magnetic flux with the quantum limited resolution k=0 b k 2 k in the so-called quantum abacus. 24 The Kitaev algorithm starts from a minimal delay τ = τ 0 and determines the most significant bit b K−1 in its first step, further proceeding with the less significant bits b K−2 , . . . , b 0 . The Fourier algorithm works backwards: 26 it starts from the maximal delay τ ∼ T 2 and first determines the least significant bit b 0 , then gradually learns more and more significant bits b 1 , b 2 , . . . , b K−1 .
Modified Kitaev-and Fourier metrological algorithms. In the present work, we use modified versions of the phase estimation protocols, which take into account the nonidealities present in actual experiments.
For brevity, we will still refer to these protocols as the Kitaev-and Fourier phase estimation algorithms. We demonstrate the superiority of these algorithms over the standard technique and show that we can beat the standard quantum limit. Instead of relying on the ideal theoretical probability function P [τ, ∆ω(Φ)] of Eq.
(1) these modified Kitaev and Fourier protocols exploit the empirical probability P p (τ, Φ), the so-called passport, which we measure by a set of Ramsey sequences at various magnetic fluxes Φ, representing the result on a discrete equidistant grid in the form . . 241 quantifying the time separation between the two π/2 rf-pulses of the Ramsey sequence. In order to increase the signal-to-noise ratio, we average over 65000 Ramsey experiments at each discrete point (τ j , Φ i ). The resulting pattern is only approximately described by Eq. (1) due to the fact that the resonator frequency changes slightly with the applied flux, thus modifying our calibration (see Methods and SI 2). In principle, one can change the working point to an even more sensitive part of the spectrum at the price of a further distortion of this pattern.
Using the qubit passport P p (τ j , Φ i ), one can pose the following metrological question: given an unknown flux Φ within some pre-chosen range, how can one estimate its value using a minimal number of Ramsey measurements? We design two metrological algorithms where the time delay τ of the Ramsey sequence serves as an adaptive parameter whose value is dynamically adjusted. In the course of operation, both our algorithms return a discrete probability distribution P(Φ i ), i ∈ I 0 , which reflects our current knowledge about the flux Φ to be measured. This probability distribution is improved in subsequent steps and shrinks to a narrow interval around the actual flux-value when running the algorithm.
Bayesian learning. The elementary building block for both our metrological algorithms is a Bayesian learning subroutine which updates the discrete flux distribution P(Φ i ) after each Ramsey measurement of the qubit state. This subroutine takes the time delay τ j between π/2 pulses as an input parameter and performs a sequence of N = 32 Ramsey measurements. Our readout scheme returns a measured variable h N which, at N 1, is equal to the empirical passport probability P p (τ j , Φ i ). At small values N , the readout variable h N is a normally distributed random variable with a mean value given by P p (τ j , Φ i ), where the variance σ 2 N = σ 2 1 /N can be directly measured, σ 2 1 ≈ 3.5 (see SI 1 for further explanations on the readout variable h N ). Next, the algorithm makes use of the measurement outcomes h N and updates the flux probability distribution with the help of Bayes' rule, Kitaev algorithm. The Kitaev-type metrological algorithm has been introduced earlier in Ref. 25 The algorithm involves K steps k = 0, . . . , K − 1 with optimized Ramsey times τ k , tolerances k , and flux index sets I k ; below, N (I) denotes the size of a discrete set I. It is initialized with a uniform discrete distribution P 0 (Φ i ) which reflects our prior ignorance of the flux to be measured. In the first step k = 0, the algorithm repeats the Bayesian learning subroutine at a zero time delay τ (0) = 0 between π/2 pulses until the probability distribution shrinks to a twice narrower interval I 1 ⊂ I 0 , i.e., N (I 1 ) = N (I 0 )/2, satisfying The flux values Φ i , i / ∈ I 1 are discarded. After completing the first step, the algorithm searches for the optimal delay τ j for the next step. The next optimal Ramsey measurement requires a larger delay τ (1) > 0 such that the passport P p (τ (1) , Φ i ), i ∈ I 1 , has the largest range: The algorithm thus sweeps over the passport data P p (τ j , Φ i ) to find the optimal delay τ (1) with maximal range ∆P (τ (1) ). Subsequently, a new distribution P 1 (Φ i∈I1 ) = N −1 (I 1 ) and P 1 (Φ i / ∈I1 ) = 0 is initialized and the algorithm proceeds to the next step by running the Bayesian learning with the new optimal delay τ 1 . After K steps, the algorithm localizes Φ within a 2 K times narrower interval I K , N (I K ) = N (I 0 )/2 K , with an error probability = 1 − . Quantum Fourier algorithm. This algorithm starts from the Ramsey measurement with an optimal time delay τ (s) ∼ T 2 . The starting delay τ (s) is a free input parameter of the algorithm. Similarly to the Kitaev algorithm, the quantum Fourier algorithm runs the Bayesian learning subroutine until the flux probability distribution P 0 (Φ i ), i ∈ I 0 , squeezes to a twice narrower subset S 1 ⊂ I 0 such that i∈S1 P 0 (Φ i ) ≥ 1 − 0 .
However, in contrast to the Kitaev algorithm, the passport function P p (τ (s) , Φ i ) is an ambiguous function of Φ i at the large delay τ (s) . As a result, S 1 is not a single interval but rather a set of n ∼ τ (s) /τ 0 disjoint narrow intervals S 1 = I 1 ∪ · · · ∪ I n of almost equal lengths, see Fig. 2. Hence, after completing the first step, the flux value is distributed among n equiprobable alternatives I i . The Fourier algorithm discriminates between these n alternatives in the next steps. First, it searches for the next optimal delay τ j , where it is possible to rule out half of the remaining alternatives in the most efficient way. At each delay τ j the algorithm splits the remaining intervals I i , i = 1, . . . , n into two approximately equal-in-size groups A = I i1 ∪ · · · ∪ I i [n/2] and B = I i [n/2]+1 ∪ · · · ∪ I in which are ordered by the passport function, P p (τ j , Φ ∈ A) > P p (τ j , Φ ∈ B). Then it finds the probability distance ∆P (τ j ) = min i∈A P p (τ j , Φ i ) − max i∈B P p (τ j , Φ i ) > 0 separating the two sets A and B. Repeating this procedure at all available delays τ j , the algorithm finds the optimal delay τ (1) with maximal ∆P (τ j ) over the discrete set of delays τ j . In the next step, the algorithm discriminates between A and B by repeating the Bayesian learning subroutine approximately [∆P (τ (1) )] −2 times and sets S 2 = A or B. Continuing in this way, the algorithm returns a single interval I out where the actual value of the flux Φ(Φ i ), i ∈ I out , is located. Fig. 2 shows how the flux distribution function P(Φ i ) develops in time during the execution of the Kitaev and Fourier algorithms.
Results. The superiority of our quantum metrological algorithms is clearly demonstrated by the scaling be- In the second step, the Fourier algorithm proceeds to a shorter delay and rules out another half of the remaining six intervals. In the next two steps the algorithm discriminates between the remaining three alternatives and ends up with the correct flux interval. The green line at the fourth step displays the probability distribution learned by the standard (classical) procedure during the same number of Ramsey measurements as was required by the quantum procedures. The distributions obtained at the step number 4 for the Kitaev and Fourier estimation algorithms and in the standard (classical) measurement are shown in the inset.
haviour of the magnetic flux resolution with the total sensing time of the flux measurement, see Fig. 3. We run each algorithm n = 25 times at every flux value Φ = Φ i , i ∈ I 0 , within the entire flux range, and find the corresponding arrays of estimated valuesΦ ji , j = 1, . . . , n. The estimateΦ =Φ P(Φ) is defined as the most likely value derived from the observed probability distribution P(Φ). For a probability distribution P i (Φ) measured at a known flux value Φ = Φ i the corresponding estimateΦ ji is a random quantity due to statistical nature of the measurement procedure. We define an aggregated resolution δΦ as an ensemble standard deviation of the random variablesΦ ji − Φ i , As follows from Fig. 3, the 5-step Kitaev procedure takes ≈ 0.05 s, which provides a value δΦ 2 ∼ (6.4 × 10 −5 Φ 0 ) 2 for the flux fluctuations, about twice larger than the actually achieved flux resolution δΦ ∼ 3 × 10 −5 Φ 0 . This suggests that 1/f flux noise has a smaller weight and another, non-magnetic decoherence mechanism is present in our device. One of the poten- Figure 3: Observed scaling behavior of the flux resolution versus total sensing time for the three different metrological procedures, Kitaev (colored circles), Fourier (black diamonds) and standard (red crosses). The Kitaev algortihm has been run with constant tolerances k = for each step k = 1, . . . , 5 and for five different values of as indicated by different colors. The Fourier algorithm has been performed with the stepdependent tolerances k = 0.182, 0.076, 0.039, 0.02, 0.01 for k = 1, . . . , 5. We show the result of the Fourier algorithm only for the final two steps, k = 4 (filled diamonds) and k = 5 (empty diamonds), running the algorithm with four different starting delays, τ (s) = 300, 320, 340 and 360 ns (all collapsed to the same data points). The phase estimation algorithms lag behind in precision at short times when compared to the standard procedure, but rapidly gain precision at longer times. The black solid line represents the scaling law for a numerical simulation of the standard procedure with a regular passport function given by Eq. (1). The crossover to the red solid line is due to the irregularity of the passport function. tial candidates derives from electron tunneling at defects inside the dielectric layer of the qubit's Josephson junctions. These fluctuating charges produce 1/f noise in the critical current and hence affect the transition frequency of the transmon atom. 27 The 1/f flux noise may become more pronounced at a larger size L of the transmon's SQUID loop as the flux-noise spectral density grows linearly with the loop size. 23  Interestingly, the non-ideality of the qubit's passport strongly affects the performance of the standard procedure as well. Its scaling behaviour δΦ std ∝ t −α exhibits a crossover in the scaling exponent α, assuming a value α ≈ 0.39 at short sensing times, while at large times α decreases to a much smaller value ≈ 0.046.
The scaling exponent 0.39 deviates from the shot noise exponent 1/2 due to the cases when the actual flux value is located near the boundaries of the flux interval Φ ∈ [Φ 1 , Φ 161 ], where the passport function P p (0, Φ) has an extremum and the scaling exponent for the standard procedure is reduced to 1/4, δΦ(t) ∝ t −1/4 . As a result, the aggregated scaling exponent of Eq. (5) is reduced below 1/2. On the other hand, the crossover to α ≈ 0.046 is a consequence of the irregularity of the passport function set by low-frequency noise fluctuations during the passport measurement. Indeed, at large sensing times, the standard procedure needs to distinguish fluxes within a narrow interval where the passport function P p (τ = 0, Φ) has a non-regular and nonmonotonic dependence on Φ. As a result, the Bayesian learning procedure fails to converge to a correct flux value. In contrast, at a larger scale of Φ, the passport function is smooth and monotonic and the standard procedure behaves properly. These arguments are indeed confirmed by a numerical simulation with a regular passport function given by Eq. (1). Importantly, both our quantum metrological algorithms are more stable than the standard procedure with respect to passport imperfections and their scaling behaviour at large sensing times coincides with the scaling behaviour resulting from a regular passport function. The quantum algorithms suffer, however, from the same irregularity problem at larger sensing times, not shown in Fig. 3.
Finally, we discuss how our metrological algorithms use the quantum resource of qubit coherence in order to acquire information about the measured flux. We quantify the quantum coherence resource spent in a given measurement by the total phase accumulation time The scaling exponent is still below the Heisenberg limit, which is a consequence of the finite dephasing time T 2 : at large time delays τ ∼ T 2 the visibility of the Ramsey interference pattern decreases, requiring more Ramsey measurements in order to learn the next bit of information.

Discussion
We have used a single transmon qubit as a magnetic-flux sensor and have implemented two quantum metrological algorithms in order to push the measurement sensitivity beyond the standard shot-noise limit. In our experiments, we utilize the coherent dynamics of the qubit as a quantum resource. We demonstrate experimentally, on the same sensor sample, that suitably modified Kitaev- (1) at different dephasing times T 2 ranging from 10 µs (red line) to 340 ns (blue line). One can clearly see that at large dephasing times the Kitaev procedure approaches the Heisenberg limit, while at smaller T 2 the scaling exponent decreases to the standard quantum limit 0.5. The observed experimental scaling behaviour shows that both Fourier and Kitaev algorithm are indeed quantum with a scaling exponent above the standard quantum limit 1/2, see the dash-dotted cyan line connecting the Kitaev (at 0.2% tolerance) data.
The transmon 18 is a capacitively-shunted split Cooper-pair box, with a Hamiltonian where E C is the charging energy E C = e 2 /2C Σ with C Σ the total capacitance (dominated by the shunting capacitor). The SQUID loop in the transmon design provides a flux-dependent effective Josephson energy E J (Φ) = E JΣ | cos(πΦ/Φ 0 )| (assuming identical junctions). The state of the device is described by a wavefunction which treats the superconducting relative phase across junctions ϕ as a quantum variable similar to a standard coordinate. In contrast to standard SQUID measurements, the flux dependence is reflected in the quantized energy levels; for the first transition this reads The readout of the qubit state is realized by a dispersive coupling of the transmon to a coplanar waveguide (CPW) resonator whose resonance frequency depends on the transmon state. This allows us to perform a non-demolition measurement of the qubit state by sending a probe pulse to the CPW right after the second π/2-pulse and collecting the resulting resonator response signal whose shape in time depends on the qubit state, see SI 1.
One can estimate the corresponding decay rates Γ wn = (520ns) −1 and Γ 1/f = (420ns) −1 , from the freeinduction decay of the qubit state at the working point of the qubit spectrum, see SI 2. If we assume that the main dephasing mechanism is due to the intrinsic magnetic flux noise of the SQUID loop ν(t) = δΦ(t), one can translate these rates into the corresponding noise spectral densities, S wn ≈ (5.9 × 10 −8 Φ 0 ) 2 /Hz and As suggested in ref. 23

Voltage-to-flux conversion
The magnetic flux threading the transmon SQUID loop is generated by a dc-current flowing through the fluxbias line located nearby the SQUID loop with the current controlled by a dc-voltage V ∈ [0.977, 1.009] V generated with an Agilent 33500B waveform generator (see SI 2). As a result, our device can also be operated as a sensitive voltmeter. The conversion from voltage values to the non-integer part of the normalized flux (Φ/Φ 0 − n), where n is an integer number, is obtained from spectroscopic measurements (Fig. 1a), and has the form Here, V 0 is the periodicity (in volts) of the CPW resonator and qubit spectra, which corresponds to the magnetic flux change by one flux quantum, and Φ tr is the residual flux trapped in the SQUID loop. Measuring the CPW resonator spectrum periodicity (see Fig. 1a inset), one finds V 0 = (12.55 ± 0.05) volts, and the trapped flux value can be found from the position of the qubit spectrum maximum ω 01 [Φ(V )] (Fig. 1a), which gives Φ tr /Φ 0 = 0.059 ± 0.004. Hence, our qubit based magnetic flux sensor measures a flux change relative to some reference value.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.