Asymptotically optimal probes for noisy interferometry via quantum annealing to criticality

Quantum annealing is explored as a resource for quantum information beyond solution of classical combinatorial problems. Envisaged as a generator of robust interferometric probes, we examine a Hamiltonian of $N\gg 1$ uniformly coupled spins subject to a transverse magnetic field. The discrete many-body problem is mapped onto dynamics of a single one-dimensional particle in a continuous potential. This reveals all the qualitative features of the ground state beyond typical mean-field or large classical spin models. It illustrates explicitly a graceful warping from an entangled unimodal to bimodal ground state in the phase transition region. The transitional “Goldilocks” probe has a component distribution of width $N^{2/3}$ and exhibits characteristics for enhanced phase estimation in a decoherent environment. In the presence of realistic local noise and collective dephasing, we find this probe state asymptotically saturates ultimate precision bounds calculated previously. By reducing the transverse field adiabatically, the Goldilocks probe is prepared in advance of the minimum gap bottleneck, allowing the annealing schedule to be terminated “early.” Adiabatic time complexity of probe preparation is shown to be linear in $N$.

Figure 1

Pseudopotential $V\left(y\right)$ from Eq. (6) that results in the large-$N$ limit from mapping the quadratic spin Hamltonian onto a 1D particle in a potential well. The evolution from a single well in region III through a critical region II to a double well in region I is apparent as the transverse field is decreased or, equivalently, as the annealing parameter $\mathrm{\Gamma }$ is swept from 1 to 0 (large inset). In region III where the transverse field is strongest, the ground state will be all spins aligned with the field along the $x$ axis. In the variable $y$, this is a Gaussian state of width $1/\sqrt{N}$. Gaussian states are the ground state of a quadratic potential. At $\mathrm{\Gamma }=1$, however, the pseudopotential of the inset figure looks like a semicircle. There is effectively no disparity because the ground state is very narrow for large $N$, and has very low probability away from $y=0$; within this locale $V\left(y\right)$ also looks quadratic. The dark (blue) contour in region II indicates the quartic form of the potential at the critical point, where the annealing parameter ${\mathrm{\Gamma }}_c=\frac{2}{3}$ or ${\gamma }_c={\mathrm{\Gamma }}_c/(1-{\mathrm{\Gamma }}_c)=2$. For $N=100$, the right-hand-side vertical sequence shows the ground state of the original spin Hamiltonian (red unbroken peaks) overlaid by the pseudopotential (blue dashed curve), with potential minima indicated by dots. For an $N=200$ ensemble, eigenstates $|{\psi }_n\rangle$ with $n\in \{0,1,2,20\}$ pass through the critical region in the uppermost four panels; lower-energy states exhibit a pitchfork bifurcation associated with the phase transition, but the highly excited $n=20$ state barely registers the structure of the potential far below (positive amplitudes in red, negative in blue).

Figure 2

In upper plot (a) energy levels $E_n$ for $n\in [0,11]$ (left vertical axis, multiple curve sequence with maxima near ${\mathrm{\Gamma }}_c$) and ground-state energy gap $\mathrm{\Delta }{E}_{20}=E_2-E_0$ (curve plotted with minimum at ${\mathrm{\Gamma }}_0$, right vertical axes) for an $N=100$ spin ensemble as the annealing parameter $\mathrm{\Gamma }$ is swept from 0 to 1 through the location of the minimum gap ${\mathrm{\Gamma }}_0$ and the critical ${\mathrm{\Gamma }}_c$ where the phase transition occurs in the thermodynamic limit $N\gg 1$. In regions I and III, energy levels are uniformly distributed $\mathrm{\Delta }E\propto 1/j$, although in region I neighboring even- and odd-numbered levels pair up; only exponentially small gaps separate them. Dashed curve indicates the energy gap in the thermodynamic limit of Eq. (14). Plot (b) shows region II landmarks: numerical results in the scale-free system are shown for the precision penalty factor (lower red curve) from Eq. (18), and ground-state gap (upper blue curve) near the critical point $a=0$ (or $\mathrm{\Gamma }=\frac{2}{3}$). It is observed that $a_F$, the point of maximum precision is to the right of the minimum gap $a_0$, and therefore ${\mathrm{\Gamma }}_F>{\mathrm{\Gamma }}_0$, also. Any annealing schedule from strong to weak transverse field will reach the point of highest precision before it encounters the minimum gap.

Figure 3

Rescaled spectrum. The eigenstates ${\varphi }_{0,1,2}$ (unbroken, dashed, dotted lines) in (a) and energy eigenvalues ${\epsilon }_{0,1,2}$ in (b) found numerically for the scale-free quartic potential from Eq. (10) with single parameter $a$. The minimum gap $a_0$ and maximum precision point $a_F$ are indicated. In the third subplot (c), blue points show the difference $\mathrm{\Delta }\mathrm{\Gamma }$ between the value of the annealing parameter at the minimum gap ${\mathrm{\Gamma }}_0$ and the $j\gg 1$ critical point ${\mathrm{\Gamma }}_c=\frac{2}{3}$ in the original discrete spin Hamiltonian of Eq. (1). The (red) unbroken curve in subplot (c) has function $0.349/j^{2/3}-0.183/j^{4/3}$ derived in Sec. 5.

Figure 4

Local vs global noise. For $N=31$ qubits, processes of collective or global or correlated dephasing (green continuous curve), individual or local or uncorrelated dephasing (blue dotted curve) and local relaxation (red dashed curve) are seen to have the same detrimental effect on precision $F$ for the annealed system. To leading order, they all produce the same characteristic precision curve (upper plot) for the full range of values of the annealing parameter $\mathrm{\Gamma }$. The inset plot shows the performance in the noiseless case (unbroken black line) compared with the three noisy scenarios, which it dwarves in region I (it approaches $F=N^2$ at $\mathrm{\Gamma }=0$). When instead the expectation value of the spin operator ${\widehat{J}}^2$ is plotted (lower plot) one observes that collective dephasing preserves the overall spin quantum number $j=j_{\text{max}}=\frac{31}{2}$, so $j(j+1)=\left(\frac{31}{2}\right)\times \left(\frac{33}{2}\right)$. Local noise, in contrast, results in drift and diffusion of the probe state into lower spin spaces. Local relaxation (red dashed line) of the GHZ-like state in region I is particularly symmetry breaking, resulting in a substantial drift into lower spin spaces. All three noise processes, however, dramatically reduce the precision of this fragile superposition state; $F$ approaches zero for all decoherence types in the weak field limit $\mathrm{\Gamma }\sim 0$. We chose $\{{\kappa }^0,{\kappa }^{\left(L\right)}\}\mapsto \{0.009,0\}$ for collective dephasing and $\{0,31\times 0.009\}$ for local noise, deliberately keeping the overall precision limit the same: $F<1/({\kappa }^0+{\kappa }^{\left(L\right)}/N)$ and to illustrate how similar the $F$ curves can be. Appendix pp4 has more detailed discussion of noise. For the local noise case, the qubits are each coupled to their own individual bosonic baths, rather than to the same bath, as in the collective noise case.

Figure 5

Numerics and analytics compared for collective dephasing. In (a), dashed curves show precision $F\left(\mathrm{\Gamma }\right)\le 1/{\kappa }^0$ for analytic ground states of regions I and III in the thermodynamic limit: states discussed in Sec. 6 and precision given in Eq. (23). Here, $N=51,101,201$ qubit ensembles are subjected to collective dephasing ${\kappa }^0=\frac{1}{200}$. Exact numerical results are unbroken curves of the same color, with the calculation proceeding as follows. First, ground states of the original spin problem are found by direct matrix diagonalization. Second, the QFI is calculated by additional diagonalization of the mixed state to which the ground state evolves under decoherence [see Eq. (C1) in the Appendix]. The discrepancy between dashed and unbroken curves in the upper figure (a) is due to the absence of (Goldilocks) region II in the thermodynamic limit (dashed curves). The lower plot (b) zooms in on the critical region for larger dephasing ${\kappa }^0=\frac{1}{50}$ and $N\in \{51,101,201,401\}$. Colored curves are still numerical results of QFI. The steep unbroken black curve in (b) gives the analytic locus of the predicted QFI maxima $\{{\mathrm{\Gamma }}_F,F\left({\mathrm{\Gamma }}_F\right)\}$ for all finite $N$, using the asymptotic results of Eqs. (19) and (20). The formula for $F\left({\mathrm{\Gamma }}_F\right)$ is only valid when the condition ${\kappa }^0{N}^2\gg 1$ is met, as it is in (b).

Figure 6

Annealing. Upper plots (a) and (b) show for $N=100$ the overlap $|\langle \psi |{\psi }_n\rangle |$ of the annealed state with the 20 lowest-energy states $|{\psi }_n\rangle$ with $n\in \{0,19\}$ subject to a linear annealing schedule of total time $\tau$, so $\mathrm{\Gamma }\left(t\right)=1-(1-{\mathrm{\Gamma }}_{\tau })t/\tau$. Odd-numbered energy eigenstates are excluded because their overlap with the annealed state is zero at all times. They have odd parity and the annealing Hamiltonian respects the even parity of the initial state at $t=0$. On the left, the annealing schedule is terminated at the critical point ${\mathrm{\Gamma }}_{\tau }={\mathrm{\Gamma }}_c$, and on the right, it is terminated some way into region I after traversing the minimum gap. When the annealing is halted in the critical region, the instantaneous ground state is still dominant across many orders of annealing time. The lower two plots (c) and (d) show the probability $P_0$ to be in the target ground state, again for an annealing cycle halted at $\mathrm{\Gamma }=\frac{2}{3}$ and 0.55. The horizontal axis is the total annealing time in the linear schedule, in units scaled by the number of qubits: $\tau /N$. The blue, green, and red curves are for systems of $25,50$, and 100 qubits, respectively; larger systems exhibit more fluctuations about the same general trend in (d). In (c), the three trend curves are identical and smooth, exhibiting no superimposed fluctuations. Annealing success for a linear schedule apparently scales linearly with ensemble size, discussed in Sec. 10.

Figure 7

Convergence to asymptotics. Vertical axes plot pure numerical factors expected to converge on ${\epsilon }_2\left(a_0\right)-{\epsilon }_0\left(a_0\right)$ in (a) and $2{\epsilon }_0\left(a_F\right)-a_F{\epsilon }_0^{\prime }\left(a_F\right)$ in (b), a rearrangement of Eqs. (11) and (20). The asymptotic predictions are 3.6519 and 1.4239, respectively, for the gap and penalty factor (blue and red dashed lines). Since these derive from the scale-free picture, they are independent of both $\mathrm{\Gamma }$ and $N$.

Figure 8

Three annealing regions exist: (I) weak field, (II) critical field, and (III) strong field. Discrete ground-state amplitudes and second-excited-state amplitudes for a system of 50 qubits $(N=2j)$ are indicated by black dots. The equivalent continuous wave-function eigenstates for the variable-mass particle are shown by upper (red) (ground-state) and lower (green) (second-excited-state) lines.

Figure 9

Global entanglement: upper plot (a) shows the analytic bounds on global geometric entanglement $\mathcal{G}$ in the thermodynamic limit (dashed green curve) as a function of $\mathrm{\Gamma }$. The unbroken green curves depict the entanglement for finite-sized ensembles up to $N=200$ (black curve). The superimposed red and (dotted) blue distributions show the amplitudes of the nearest spin-coherent state (red) to the ground state of the annealed system [blue (dotted)] in the original spin problem. This spin-coherent state is also the nearest (or highest fidelity) separable state. Finding this state is crucial, due to $\mathcal{G}$ being the negative logarithm of the square of the overlap between these two states. When the ground state centered on $y=0$ bifurcates into two lobes as it passes from region III to region I the nearest spin-coherent state (having an approximately Gaussian distribution) can only track one of the lobes; its mean $y_{\alpha }=cos\alpha$ starts to move with decreasing $\mathrm{\Gamma }<\frac{2}{3}$. As $\mathrm{\Gamma }$ approaches 0, the entanglement asymptotes to $\mathcal{G}=1$, the entanglement of a GHZ state. The lower graph (b) plots the entanglement $\mathcal{G}$ more closely in region II for a system of $N=100$ qubits in the scale-free setting. Maximum entanglement is seen to occur in the vicinity of the minimum gap, but this maximum will be closer to $a=0$ or $\mathrm{\Gamma }={\mathrm{\Gamma }}_c$ for larger $N$. As $N$ increases the maximum width of a spin-coherent state scales as $N^{-1/6}$ in the $z$ variable. Thus, maximum entanglement grows slowly as $\frac{1}{6}lnN$. The red dotted curve plots the locus of the mean $y_{\alpha }$ of the nearest spin-coherent state. Apparently, as $a$ or $\mathrm{\Gamma }$ decreases the ground state has already begun to bifurcate before maximum entanglement is reached.