Quantum speedup at zero temperature via coherent catalysis

It is known that secondary nonstoquastic drivers may offer speedups or catalysis in some models of adiabatic quantum computation accompanying the more typical transverse field driver. Their combined intent is to raze potential barriers to zero during adiabatic evolution from a false vacuum to a true minimum; first-order phase transitions are softened into second-order transitions. We move beyond mean-field analysis to a fully quantum model of a spin ensemble undergoing adiabatic evolution in which the spins are mapped to a variable mass particle in a continuous one-dimensional potential. We demonstrate that the necessary criterion for enhanced mobility or “speedup” across potential barriers is actually a quantum form of the Rayleigh criterion. Quantum catalysis is exhibited in models where previously thought not possible, when barriers cannot be eliminated. For the 3-spin model with a secondary antiferromagnetic driver, catalyzed time complexity scales between linear and quadratic with the number of qubits. As a corollary, we identify a useful resonance criterion for quantum phase transition that differs from the classical one, but converges on it, in the thermodynamic limit.

Figure 1

Pairing of energies during transition from the unimodal to bimodal ground state. Here ${\xi }_1$ is a displacement variable and $\beta$ a “width” variable. This important transition we call the quantum Rayleigh limit: ${\xi }_1\sim \beta$, derived later in Eq. (5). As plotted, adjacent even and odd eigenvalues $E_k^{\pm }$ of the piecewise-parabolic potential are seen to “pair up,” to become doubly degenerate, as the scaled separation of the wells increases beyond the order of the well widths, i.e., ${\xi }_1\gg \beta$ (above for the case $\beta =1$). In this limit (blue-shaded region) the energies are well below the barrier height (dashed line) and the intrapair spectral gap decreases exponentially with the barrier height $V_0$. A sketch of the ground-state wave function is shown (upper middle) in the limit of well-separated wells. For ${\xi }_1<\beta$, violating the Rayleigh criterion, the wells coalesce and the vacuum energy $E_0^{+}$ (red) and other eigenenergies subsume the barrier. At the leftmost edge the two wells merge completely, and eigenvalues revert to those of a simple harmonic oscillator: $(n+1/2)\hslash \omega$.

Figure 2

Three quantum transport mechanisms exist in a double well during annealing. The potential barrier (dark blue curve) has height $V_0$ and the vacuum energy is $E_0^{+}$ (red horizontal line). The inverted parabola centered at the origin is stitched to two parabolic wells centered on $\pm {\xi }_1$ with characteristic width $\beta$. The join location is $\xi =\pm {\xi }_1/(1+{\beta }^4)$ (vertical dashed lines in all subplots). The upper panel indicates two well-understood mechanisms of quantum tunneling (red dashed line) through the barrier and classical activation over the barrier (black dashed line) following absorption by a thermal photon from the environment. The ground state is shown as a gray profile in all three panels. Tunneling and thermal activation are possible transport mechanisms when the wells are far apart, ${\xi }_1\gg \beta$, and the barrier is larger than the vacuum energy, $V_0>E_0^{+}$. There is, however, another transport mechanism, vacuum delocalization, that comes into play when the width of the ground state in one isolated well approaches the well separation, i.e., $\beta \approx {\xi }_1$, perhaps initiated by some external catalysis. This “Rayleigh limit” may occur when the vacuum energy subsumes the barrier, as in the middle panel. While sufficient for delocalization, this is by no means a necessary condition—as the lower panel demonstrates. The potential in this lower panel has values $(\alpha ,\beta )\mapsto (2,2)$, as compared with $\mapsto (3,1)$ in the upper panel, and $\mapsto (1.08,1.08)$ in the middle panel.

Figure 3

Contour plot of isogaps (constant spectral gap ${\mathrm{\Delta }}_{01}$) in $\{{\xi }_1,\beta \}$ parameter space for the potential of Sec. 3. Rayleigh limit is indicated by magenta line. As in the previous figure, the effective width of the wells vs the width of the top of the barrier is characterized by the ratio $\beta ={\sigma }_1/{\sigma }_{*}$ from Eqs. (A2). The distance between the well minima is $2{\xi }_1$ in the same scale-free units. Gap ${\mathrm{\Delta }}_{01}$ is exponentially small when the ground state is split into two localized components (lower right sketch), one in each well; ${\xi }_1>\beta$. If these two parameters become comparable the gap approaches $\hslash {\omega }_1$ as the wells effectively merge (upper right sketch). For a bulk unimodal state of effective width $L$ the energies and gaps will scale $\propto L^{-2}$ (particle in a box). For fixed well separation ${\xi }_1$ the magenta curve traces $\beta$ values associated with a maximum gap, i.e., where the indicated isogap contours are vertical. This line asymptotes to $\beta \sim 0.82{\xi }_1$. For larger $\beta$ (above the magenta line) ${\mathrm{\Delta }}_{01}$ remains large but the well energies and gaps decrease as $1/{\beta }^2$, no longer exponentially.

Figure 4

An asymmetric well may also be modeled as piecewise parabolic, with two location and two width parameters, $\{{\xi }_{1,2},{\beta }_{1,2}\}$, respectively. Here we have chosen different width parameters $\{{\beta }_1,{\beta }_2\}\mapsto \{1,3^{-1/4}\}$ for all plots, and equal location parameters ${\xi }_1={\xi }_2\approx 1.25$. These values correspond to the optimal catalysis of the quantum 3-spin; see Eq. (22). To identify the point of phase transition at the barrier summit the vacuum energy $\epsilon$ of all three potentials must coincide: the asymmetric potential and both symmetric wells from which it is composed (vacuum energy level depicted by red horizontal lines). Near the barrier summit the potential is an inverted parabola; the ground state has identical analytical form in all three cases (upper plots): a standing wave formed from right- and left-moving parabolic cylinder functions (see Appendix pp1 for further discussion). Without this resonance condition we cannot assume that the ground state represents a system at the point of phase transition. Figure 19 illustrates the different phenomenon of resonance tunneling for slow phase transitions where neither well approaches the Rayleigh limit and the barrier greatly exceeds the vacuum energy.

Figure 5

In the quantum 3-spin model, increasing the antiferromagnetic coupling governed by $(1-\kappa )$ from Eq. (7) “softens” the discontinuous or first-order phase transition occurring for the external magnetic field parameter $\mathrm{\Gamma }=0.565$, to approach a continuous or second-order transition nearby. This is illustrated above for $n=100$ spins. For $\kappa =1$ the discontinuity in the ground state [amplitudes $\psi \left(z\right)$ on the vertical axis] is quite apparent as it tunnels from the paramagnetic phase centered on $z=m/j=0$ to the ferromagnetic phase near $z=+1$. For $\kappa =0.1$, this transition has been “smeared out” by the contribution of the nonstoquastic driver $+{\widehat{J}}_x^2$; the state distribution changes continuously with $\mathrm{\Gamma }$ in the second plot.

Figure 6

Zero-temperature energy surfaces $V\left(z\right)$ associated with quantum annealing as a transverse field $\mathrm{\Gamma }$ is lowered adiabatically (upper row of 2D plots provides an overhead plan view): Coordinate $z=m/j\in [-1,1]$ is the ratio of the magnetic quantum number $m$ to total spin $j$. For an $n$-qubit ensemble confined to the fully symmetric subspace, the total spin $j=n/2$. Loci of the maxima (dashed red line) and minima (unbroken red line) are indicated, as are births of new maximum-minimum pairs (red dots), and second-order phase transitions where a minimum becomes a maximum (black dots). These latter points are associated with lines of zero curvature, $V^{″}\left(z\right)=0$. As $\mathrm{\Gamma }$ is reduced, the ground state evolves in (a) and (b) by a first-order phase transition connecting $z=0$ via tunneling through an intervening barrier (dark blue contour) to reach the global minimum at $z=1$. The blue and cyan contours trace out the potential wells when they have equal depth, as defines the classical first-order transition. (In the 2D plan view, the blue transition line intersects the red dashed line associated with a potential maximum, a signature of barrier penetration.) To contrast, case (c) illustrates the Lipkin-Meshkov-Glick (LMG) model [12, 13], in which the ground state smoothly evolves from being localized at $z=0$ to $z=\pm 1$, bifurcating continuously near the critical point ${\mathrm{\Gamma }}_c=2/3$ (green line). The “gentler” phase transition here is described as second order or continuous; the green contour intersects no intervening $V\left(z\right)$ maximum, and there is no barrier penetration. Cases (a) and (b) present the $p$-spin model for $p=3$ which, at least classically, presents an unavoidable first-order phase transition; the barrier cannot be fully suppressed to zero. Case (b) shows the modified potential produced by the presence of the nonstoquastic catalyst term in the $p$-spin Hamiltonian, a transverse ferromagnetic coupling of strength $(1-\kappa )$, as described in Sec. 5. The catalysis softens the transition, bringing the first- and second-order transition points in close proximity (red and black dots of top middle figure) and effecting partial barrier suppression.

Figure 7

Any annealing schedule that maps $\{\mathrm{\Gamma },\kappa \}$ values $\{1,1\}\mapsto \{0,1\}$ will unavoidably traverse the minimum gap region that corresponds to the phase transition of the quantum 3-spin model, Eq. (7). The phase transition ridge may not be circumvented, as is quite apparent in the upper contour plot of the inverse gap $1/\mathrm{\Delta }$ for $n=50$ ($j=25$). The ridge extends throughout the full parameter range $\kappa \in [0,1]$. Near the line $\kappa =1$ the transition is first order; the associated minimum gap is always exponentially small in $j$. However, for optimized control parameters $\{{\mathrm{\Gamma }}_c,{\kappa }_c\}=\{0.598,0.479\}$ (denoted by a white star) the ridge defining the phase transition has a saddle, a “maximum minimum gap.” Optimized annealing schedules that minimize computation time should pass near this point. In fact, for $\kappa \le {\kappa }_c$ as $j\gg 1$ a polynomially small minimum gap is always possible, even though a classical analysis of the potential landscape alone would seemingly forbid this. The lower plot shows the $\{{\mathrm{\Gamma }}_c,{\kappa }_c\}$ parameter pairs (stars) for different labeled $17.5\le j\le 95$. (For $j\le 17$, optimal ${\kappa }_c=1$.) The orange and blue curves describe respectively the birth of the second potential minimum $V^{″}\left(z_1\right)=0$ and the point at which both wells are of equal depth, $V\left(z_1\right)=V\left(0\right)$. The green line corresponds to the second-order phase transition at $z=0$. It is “hidden” in the sense that any adiabatic annealing schedule progressing from right ($\mathrm{\Gamma }=1$) to left sides will initially encounter the first-order transition from the $z=0$ minimum to $z_1>0$; the green curve never crosses the blue-orange bounded region, for all $\kappa$.

Figure 8

For direct comparison, here are plotted the spectral gap “landscape” for both systems, as a function of the control parameters $\mathrm{\Gamma }$ (transverse field) and $\kappa$ (antiferromagnetic driver). The upper plot depicts, for $n=80$ spins, the 3-spin ensemble. The lower figure is the equivalent analytical model: a particle of variable mass moving in a one-dimensional continuous potential well. This 1D model is seen to exhibit the same qualitative features as the original spin ensemble (upper plot), except for an infinite-mass singularity at $z=0$ coincident with the hidden second-order phase transition (white dashed curve in lower plot); see Eq. (19). The saddle point of the phase transition ridge for the original 80-spin system is indicated (in both plots) by a white star marker, and with a cyan marker for the continuous model. Ground-state wave functions associated with both models are shown in Fig. 11.

Figure 9

The phase transition ridge features a saddle point, or “maximum minimum gap,” more clearly visible in 3 dimensions when plotted against control variables $\kappa$ and $\mathrm{\Gamma }$. The energy gap of this saddle point scales polynomially with system size $j$. At larger $\kappa$ (reducing the influence of the nonstoquastic driver) the gap size shrinks to be exponentially small in $j$. For very small $\kappa$ the gap again quickly closes from polynomial to exponential or factorial; this anomalous gap was first discussed in Ref. [25].

Figure 10

For the quantum 3-spin model, the Hamiltonian contribution of the transverse antiferromagnetic driver $+(1-\kappa ){\widehat{J}}_x^2$ is illustrated near the classical first-order phase transition (wells are of equal depth). Control parameters $\kappa$ (indicated) and $\mathrm{\Gamma }$ (not shown) and are chosen to fulfill this condition, and as $\kappa \mapsto 0$ the contribution of the nonstoquastic driver increases, lowering the barrier $V_0$, and reducing the separation of the potential wells $z_1$. The paramagnetic well is centered on the origin (unbroken white line). The dashed white line describes the ferromagnetic minimum at $z=z_1$. Asymmetry of the wells is apparent even when they are of equal depth, as above.

Figure 11

The Rayleigh limit is clearly violated at optimal catalysis for both the original discrete spin system and the continuous model, i.e., for the widest minimum gap ${\mathrm{\Delta }}_{01}$, here illustrated for $n=80$ qubits. Optimal control parameters $\{{\mathrm{\Gamma }}_c,{\kappa }_c\}$ were found for the spin system by numerical search using a truncated Newton method. The ground state $\mathrm{\Psi }\left(z\right)$ (blue and dashed red curves) no longer has isolated components in each well; they are largely coalesced. Note the asymmetry, not only of the potential well $V\left(z\right)$ (gray filled curve), but of the wave function, biased towards the ferromagnetic well (centered on $z_1$).

Figure 12

The spectral gap ${\mathrm{\Delta }}_{01}$ in the 3-spin is formed by multiplying the $O\left(1\right)$ gap in scale-free coordinates by $\hslash {\omega }_{*}$, which produces ${\mathrm{\Delta }}_{01}\sim O(1/j^2)$. To find the numerical coefficient we may apply the delocalization resonance condition of Sec. 4 to produce this curve of the gap (in arbitrary units) as a function of the Rayleigh coefficient $\alpha$. It features a maximum at $\alpha \approx 1.67$, corresponding to the saddle of the phase transition.

Figure 13

Upper plot collects numerical results for the nonstoquastic control parameter ${\kappa }_c$ of 3-spin system for $4\le j\le 196$. The optimized parameter values asymptote to ${\kappa }_c\propto 1/\sqrt{j}$, Eq. (23). For systems of $j\le 17$ apparently the ground state is always delocalized over the barrier, because the effective $\hslash =1/j$ (and therefore vacuum energy) is large. In these cases no catalysis (no barrier suppression) is necessary to produce a polynomial gap: ${\kappa }_c=1$. The lower plot shows the scaling of the minimum gap, also with system size. Even though it is forbidden from the perspective of classical phase transition theory, a polynomial gap can be maintained in the quantum 3-spin model by suppressing the potential barrier to an optimal nonzero height $V_0\propto {\kappa }_c^4$. A partially lowered barrier permits the two well components to merge; at the Rayleigh limit they become a unimodal state, with polynomial spectral gap. Without any catalysis the minimum gap will quickly close at an exponential rate (magenta line), the crossover again indicated at $j\approx 17$. The exponential scaling of the uncatalyzed first-order transition asymptotes to $\mathrm{\Delta }\sim exp\{-0.175j\}$, as documented in Ref. [5]. The optimal gap (black dots) eventually converges on the asymptote (cyan line) predicted by the continuous model, Eq. (24). The rate of gap closure is always faster than that of the Lipkin-Meshkov-Glick (2-spin) model [12, 13] ${\mathrm{\Delta }}_c\propto j^{-4/3}$, a continuous phase transition. Its scaling is indicated in the lower plot by a dashed line.

Figure 14

The algorithm due to Dijkstra is employed on a graph produced by rasterizing the control landscape of Fig. 7, here the ensemble size $j=40$. Contours here are also of $log1/{\mathrm{\Delta }}_{01}$, the cost of moving between adjacent nodes $\propto 1/{\mathrm{\Delta }}_{01}^2$. Darker-shaded regions may therefore be more rapidly traversed. Movement costs along graph edges, between pixels, are defined by Eq. (25). The green line represents an optimal adiabatic contour $C^{*}(\mathrm{\Gamma },\kappa )$ from the initialization parameters $\{\mathrm{\Gamma },\kappa \}=\{1,1\}$ (green dot) to the final values $\{0,1\}$ (red dot). The blue pixels of the upper plot represent the phase transition ridge, with the saddle indicated by a white star marker. One might imagine an explorer journeying south from his home in the northeast corner to cross a river close to the shallowest point before heading to his destination in the northwest. There are different anisotropic frictional movement costs for going south $(1-\mathrm{\Gamma })\left(\delta \kappa \right)$ versus west, $(2-\kappa )\left(\delta \mathrm{\Gamma }\right)$, independently of the node cost. This explains the zero friction path traversed due south from $\{1,1\}$ to $\sim \{1,0.3\}$ that then turns westward towards the saddle marker. On the lower plot is $t^{*}$, the cumulative contribution to the total adiabatic time. Unexplored regions are left uncolored in the lower plot. Admittedly, there will be error in the sampling of the phase transition terrain (the ridge is basically a delta function) which might be improved by adaptive sampling of regions with steep gradients.

Figure 15

In $\{\kappa ,\gamma \}$ parameter space, the quantum domain exists between the birth of the ferromagnetic well in the potential at ${\gamma }_0$ and the point of the classical first-order phase transition ${\gamma }_c$, where the potential wells have equal depth. Also shown is the hidden second-order transition ${\gamma }_2=2(1-\kappa )$, where the paramagnetic minimum becomes a maximum (dashed line). Between ${\gamma }_c$ and ${\gamma }_0$ appears the true quantum phase transition at ${\gamma }_{*}$. Scaling of ${\gamma }_0-{\gamma }_{*}$ is observed between ${\kappa }^2$ and ${\kappa }^3$; a fit line for ${\kappa }^{2.75}$ is indicated on a logarithmic scale in the lower plot (red line).

Figure 16

A light at the end of the tunnel: Subject to a full quantum analysis, the phase portrait for the antiferromagnetically driven 3-spin must be redrawn. The boundary between paramagnetic phase (PP) and ferromagnetic phase (FP) in the space of control parameters $\mathrm{\Gamma }$ and $\kappa$ is no longer first order, and admits a softer transition. This “polynomial back door” of order $1/\sqrt{j}$ bounded by the saddle point (star marker) circumvents the exponential slowdown in time-to-solution associated with tunneling; the annealing proceeds in polynomial time.

Figure 17

In the left column the ground state ${\varphi }^{+}$ valid at the summit $\xi =0$, a Kummer function from Eq. (A6) (red curve), is joined to the parabolic cylinder function of Eq. (A8) (black dashed curve) valid within the well. The area under the correct piecewise solution for ${\varphi }^{+}$ is shaded blue. The barrier height $V_0$ is proportional to the harmonic mean of the squared frequencies, Eq. (3). The particular parabolic cylinder function is chosen that decays to zero at $\xi \sim \infty$. Three cases are considered in turn, vacuum energies $E^{+}$ below, equal to, and above the barrier of height $V_0$, indicated in the right column, with rows corresponding to $\{{\xi }_1,\beta \}\mapsto \{2.5,0.5\},\{1.07,1\},\{0.5,1\}$ values. Recall that $\pm {\xi }_1$ is the location of each well minimum, the ratio of well width to barrier width is $\beta ={\sigma }_1/{\sigma }_{*}=\sqrt{{\omega }_{*}/{\omega }_1}={[V^{″}(\xi =0)/V^{″}(\xi ={\xi }_1)]}^{1/4}$, and $\pm {\xi }_1/(1+{\beta }^4)$ are the “seams.”

Figure 18

Mean spin for the 3-spin model has components $\langle \vec{J}\rangle =\langle \mathrm{\Psi }|\{{\widehat{J}}_x,{\widehat{J}}_y,{\widehat{J}}_z\}|\mathrm{\Psi }\rangle$, normalized by the principal quantum number $j$. In polar coordinates the magnitude $r=|\langle \vec{J}\rangle |/j$ and angle $\theta =arctan\{\langle {\widehat{J}}_z\rangle /\langle {\widehat{J}}_x\rangle \}$. The uncertainty is the standard deviation (red curves) calculated from the quantum variance, $j\mathrm{\Delta }r=\sqrt{{\mathrm{\Delta }}^2{\widehat{J}}_x+{\mathrm{\Delta }}^2{\widehat{J}}_y+{\mathrm{\Delta }}^2{\widehat{J}}_z}$. Dashed quarter circles in the polar plots correspond to uncertainty of a classical spin (a spin coherent state), constant at $1/\sqrt{j}$, in these units. Most of the quantum behavior is confined to a small parameter region $\mathrm{\Delta }\mathrm{\Gamma }$ close to the classical first-order phase transition (PT) value, ${\mathrm{\Gamma }}_c$. Outside the region bounded by orange and blue lines the system behaves as a large rotating spin with $r=1$, and $\mathrm{\Delta }r=1/\sqrt{j}$. Within this “quantum” boundary, however, the vector is quite nonclassical: $r<1$ and $\mathrm{\Delta }r>1/\sqrt{j}$. The orange and blue boundary lines in the polar plot correspond to the birth of the second potential minimum and the classical phase transition, respectively. At $j=40$ the appearance of the minimum spectral gap (green line) occurs at an angle almost exactly bisecting the quantum sector. The associated angle coincides with maximum uncertainty, $\mathrm{\Delta }r$. Within the boundary the spin vector describes almost a straight line chord (magenta) perpendicular to the minimum gap “event” angle (green line). The phase transition is seen to be “softer” at lower spin number, e.g., $j=10$ (cyan curves of middle column). The $j=\infty$ line (black dashed curves of middle column) tracks the global minimum of $V\left(z\right)$ exactly; this is the thermodynamic limit where the vacuum (kinetic) energy vanishes. The rightmost plots illustrates the shape of the double well for $\kappa \mapsto \{1.0,0.4\}$ at the birth of the second minimum (orange) and at the classical phase transition (blue) and for the minimum spectral gap in the case $j=40$ (green). Observe that at the minimum gap the potential is not symmetric, nor the wells of equal depth.

Figure 19

Asymmetric double well: Here is illustrated the case that the left well is outside the Rayleigh boundary: $\{{\xi }_1,{\beta }_1\}\mapsto \{3.0,1.5\}$ (white circle marker). Then one may employ WKB-like methods; the gap can become so small on resonance to be dictated by the potential shape and structure away from the parabolic extrema [20]. In such a case the piecewise-parabolic model loses its generality; errors or simplifications in the description of the potential have greater magnitude than the spectral gap calculated via this potential. The light-shaded channel identifies the resonance condition for $\{{\xi }_1,{\beta }_1,{\xi }_2,{\beta }_2\}$: an exponentially small minimum gap where one can expect a first-order phase transition via slow tunneling.

Figure 20

The Lipkin-Meshkov-Glick (LMG) model with transverse and longitudinal field parameters ${\mathrm{\Gamma }}_x,{\mathrm{\Gamma }}_z$, respectively, exhibits the relatively gentle second-order phase transition we explored in Ref. [22] (lower plot). It occurs for an annealing schedule that follows the line of ${\mathrm{\Gamma }}_z=0$ from ${\mathrm{\Gamma }}_x>2/3$ to ${\mathrm{\Gamma }}_x<2/3$ (direction of red arrow). The minimum gap ${\mathrm{\Delta }}_{02}$ near ${\mathrm{\Gamma }}_z=0,{\mathrm{\Gamma }}_x=2/3$ is polynomial in $n$; transitions between the ground and first excited state ${\mathrm{\Delta }}_{01}$ are forbidden by parity along ${\mathrm{\Gamma }}_z=0$. In contrast, an annealing schedule for which ${\mathrm{\Gamma }}_z\ne 0$ will not respect parity and the relevant gap is ${\mathrm{\Delta }}_{01}$ (upper plot). A schedule crossing this zero longitudinal field line (direction of white arrow) must undergo a first-order phase transition if ${\mathrm{\Gamma }}_x\lesssim 2/3$. Gap ${\mathrm{\Delta }}_{01}$ is then exponentially small when crossing ${\mathrm{\Gamma }}_z=0$. The landscape above is for a $j=25$ spin ensemble.

Figure 21

For both the Lipkin-Meshkov-Glick (LMG; red curve) and quantum 3-spin model (blue curve), increasing the separation of the well minima $z_1$ leads to exponentially small minimum gaps, which occur along the “ridge” of the phase transition (contour plots to the right show these ridges in parameter space and the direction along which they are traversed as $z_1$ is increased). The 3-spin model is distinctive in that the minimum gap (blue) goes through a maximum in $z_1$, indicating a saddle point of “optimal catalysis” (white star marker). Exploring the ridge in the reversed direction now, from top left to bottom right (lower right panel), the well separation continues to decrease with decreasing $\kappa$, but the inverse mass $1/m\propto {\kappa }^2$. There is apparently a competition between the quantum “particle” becoming more confined at lower $\kappa$, but heavier at the same time. The increasing mass eventually wins at very low $\kappa$, causing the spectral gap to shrink again.

Figure 22

For longitudinal field ${\mathrm{\Gamma }}_z\mapsto \{0,+0.05,+0.25\}$ the ground-state amplitude density (found numerically) and potential extrema (derived analytically) are shown in the left and right columns, respectively. For the right column, the locus of the potential maxima in coordinate space $z$ is indicated (dashed red line), with the exterior minima shown in blue (shallow) and green (deep). As the barrier maximum is lowered with increasing ${\mathrm{\Gamma }}_x$, for a symmetry-breaking ${\mathrm{\Gamma }}_z>0$ one of the minima collides with the now off-center maximum. Only the green minimum remains, moving towards the origin $z=0$ as ${\mathrm{\Gamma }}_x$ is increased further. Changing ${\mathrm{\Gamma }}_z$ to $-{\mathrm{\Gamma }}_z$ just reflects the diagrams through the origin (top to bottom). In the lower figure then the minimum at $z\approx -1$ would become the true minimum, and the (now false) ground state at $z=+1$ will tunnel through the intervening barrier (red dashed maximum line), unless the barrier is small enough, Eq. (D3), for vacuum delocalization to occur.