Grover implementation with a particle in a Box

Our system is one quantum particle confined to a square with hard outer walls. The search candidates are sixteen orthonormal spatial wave packets: four along each axis. They form a sixteen-dimensional state space within one particle’s wavefunction.

Take four sine modes along each axis. On a box of side \(L\), the one-dimensional modes are \(u_n(x)=\sqrt{2/L}\sin(n\pi x/L)\), \(n=1,\ldots,4\). Combine them into four packets \(f_x\), each with its main peak near the center of a different quarter of the interval. Products of two such packets give our \(4\times4\) basis:

The integers \(x,y\) label packets; the capital \(X,Y\) denote continuous physical coordinates. Each coefficient \(c_{xy}\) is the amplitude of one logical search state. Orthonormality makes \(\int|\psi|^2\,\mathrm dX\,\mathrm dY=\sum_{x,y}|c_{xy}|^2\).

The inner grid lines are labels, not additional walls. The packets overlap in space while remaining orthogonal as functions. Consequently, the probability of the logical outcome \(|x,y\rangle\) is \(|c_{xy}|^2\), whereas the probability of finding the particle inside the corresponding drawn square is the integral of \(|\psi|^2\) over that square. The two readouts are different.

We start in the lower-left packet, \(|i\rangle=|0,0\rangle\). In the illustrated run the target is the upper-right packet, \(|w\rangle=|3,3\rangle\). Choosing that target programs the demo’s phase oracle; the search itself only acts through the oracle.

A measurement that distinguishes the packet modes would read out the logical search result. A simple position measurement reads out position instead. This distinction matters when comparing a grid percentage with the fraction of tracers inside a square.

Inside the square, set the potential to zero and keep the hard walls. The Hamiltonian and sine-mode energies are

Here \(n\) and \(\ell\) are positive integer mode numbers and \(m\) is the particle mass. Each sine coefficient simply acquires its own phase, \(b_{n\ell}(t)=e^{-iE_{n\ell}t/\hbar}b_{n\ell}(0)\). The reconstruction in the packet basis can look complicated, but its evolution is known exactly.

For four packets per axis, wait \(T=\pi\hbar/(4E_1)\). The first four one-axis energy phases are \((z,-1,z,1)\), with \(z=e^{-i\pi/4}\). In our packet basis the resulting one-dimensional matrix is

Every entry of \(A\) has magnitude \(1/4\). A localized input therefore evolves into equal logical probabilities, with definite relative phases. The box itself performs balanced preparation and mixing.

Undoing the mixer by waiting forwards

The propagator \(A=U_{\mathrm{box}}(T)\) is not its own inverse. However, the full box spectrum revives at \(T_{\mathrm{rev}}=2\pi\hbar/E_1=8T\). Every mode then returns with exactly its original phase. We can implement \(A^\dagger\) with positive-time free evolution:

The inverse really takes seven times as long as the forward mixer. No time reversal is needed. At 1× playback, \(T\) is 2.4 seconds and the inverse interval is 16.8 seconds.

Assemble the search round

Preparation gives \(|s_A\rangle=A|i\rangle\). We need two selective phase operations: the oracle \(O_w=I-2|w\rangle\langle w|\), which flips the marked mode, and the reference \(S_i=I-2|i\rangle\langle i|\), which flips the known input mode. Undo preparation, apply the reference flip, and prepare again:

A round applies \(O_w\) followed by \(M_A\). In time order: oracle → wait \(7T\) → input phase flip → wait \(T\). Preparation is performed once, before the repeated rounds. This is amplitude amplification about the actual prepared state, even though its coefficients are complex. For the reflection geometry and sign convention, see the refresher’s gate construction. [Amplitude amplification]

At complete-round boundaries, the marked-mode probability rises from 6.25% to 47.27%, 90.84%, and 96.13%. Stop after three rounds in this example: a fourth drops it to 58.17%. The wave’s intermediate phases and density come from the box dynamics, while the endpoint probabilities follow the usual Grover amplification curve.

Explore the free box → Choose the initial packet and goal directly on the wave, or keep the illustrated defaults. Use Apply next operation to watch preparation and the much longer inverse wait. Keep phase colors on to see the interference that redistributes the wave.

The query advantage and the encoding’s physical resources are separate questions. Representing \(N\) candidates by \(N\) spatial modes does not by itself provide a scalable replacement for a \(\log_2N\)-qubit device: preparing, controlling, resolving, and measuring those modes also requires resources.

During free evolution, the wave determines the scalar Schrödinger probability current. The arrows show this current; yellow tracers are advected with the local transport velocity \(\mathbf v=\mathbf j/\rho\):

The current obeys the continuity equation, \(\partial_t\rho+\nabla\cdot\mathbf j=0\): probability leaves one region by flowing into another. The yellow dots are numerical tracers sampled from \(\rho=|\psi|^2\). Their motion helps us track how that density moves and changes shape. The viewer uses 25,000 small tracers with short trails, so local flow remains readable. The tracer count controls the detail of this visual sampling; the wave, current, and search probabilities are computed independently of it.

The phase pulses need a different transport rule

Free evolution supplies the mixing intervals. The oracle and reference are ideal mode-selective pulses, during which we omit the kinetic Hamiltonian. Each pulse rotates one packet coefficient through half a turn:

Here \(\tau\) is the pulse duration. The logical probabilities stay fixed during a pulse, but spatial density can change because the packets overlap and interfere. A mode projector is therefore not equivalent to applying a local potential inside one drawn square.

For these nonlocal pulses, the scalar demo chooses \(\mathbf j=\nabla\chi\), with \(\nabla^2\chi=-\partial_t\rho\) and zero normal current at the walls. Moving the dots with \(\mathbf v=\mathbf j/\rho\) transports the changing density consistently. This is the curl-free current chosen for the ideal pulse. It satisfies the same continuity equation as the evolving density.

We can extend the search by giving the particle spin \(\tfrac12\). Every spatial packet can now carry an up and a down amplitude. Our \(4\times4\) grid encodes 32 joint states \(|x,y,\sigma\rangle\), and the wave becomes a two-component spinor:

Mix position with the same free-box propagator and mix spin with a uniform rotation. With \(R_y(\alpha)=e^{-i\alpha\sigma_y/2}\), where \(\sigma_y\) is the Pauli y matrix, preparation and its inverse are

The spin rotation is an imposed spin-only drive. The spatial motion during mixing is still free; the entire operation also includes this drive. The oracle flips one joint position–spin mode, and \(S_i\) flips the chosen input position–spin mode. The reflection argument is unchanged.

Preparation gives \(1/32\) in every joint mode. The first recommended stop is now four rounds, at approximately 99.9182% marked-mode probability. A position measurement alone does not distinguish up from down. The demo therefore separates joint-mode probabilities from the spatial probability inside the goal box, which includes both spin components.

Spin changes the current, too

Spin introduces an additional contribution to the probability current, which can change the current’s circulation while leaving the density evolution unchanged. The position current used during free evolution with the uniform spin drive includes the convective term and this spin contribution:

In the planar model, the spin term is \((\hbar/2m)(\partial_YM_z,-\partial_XM_z)\), with \(M_z=|\psi_\uparrow|^2-|\psi_\downarrow|^2\). It is divergence-free: it can add circulation and bend the tracer paths without changing the density’s continuity equation. After a selective oracle, position and spin can become entangled. The tracers continue to sample the total density, with both spin components contributing to the current. [Pauli current]

Total · spin colors uses cyan and magenta for opposite local z polarizations. A spinor has no single scalar phase, so that total view has a different legend from the individual up/down phase views. Switch the current arrows between total, transport, and spin current to see which part produces which flow.

Explore the spin version → Select a joint goal, run four rounds, and compare the spin-resolved result with the total spatial density.

To separate the spatial encoding from the box’s natural evolution, we also built a controlled model, MultiRegion. It applies \(H=H_1^{\otimes4}\) to the sixteen packet coefficients, where \(H_1\) is the single-qubit Hadamard. The refresher derives this gate construction. Begin with \(\varphi_{00}\), prepare with \(H\), mark \(\varphi_{33}\), then apply \(H^\dagger\), \(S_i\), \(H\). Its gate endpoints reproduce the ordinary four-qubit search in this spatial basis.

To show a continuously moving wave between gates, this model uses an ideal coupling Hamiltonian over a mixing interval of duration \(\tau\). We write it as \(\mathcal H_{\mathrm{mix}}\) to distinguish it from the gate \(H\):

At \(p=1\) this gives \(H\) exactly. The inverse interval follows \(U(-p)\): the endpoint is again \(H\), but the continuous path runs with the opposite coupling. The sine modes give us a spatial basis here; their free kinetic-energy phases are not added during these controlled gates.

The oracle and reference use the mode-selective phase pulses described above. All operations in this comparison model are imposed couplings; free kinetic evolution is omitted.

The yellow dots use the gradient-current rule described in the probability-flow section throughout these controlled gates. Comparing this view with the free box reveals how two different continuous dynamics can produce the same marked-mode probabilities at complete rounds.

Explore the controlled wave model → Start with phase colors on. Step through the first oracle and compare the changing density with the unchanged marked-mode percentage.

Start with the free-box simulation to try the implementation described here. The spin and controlled-wave versions let you compare different dynamics. Ordinary columns accompany the separate refresher and provide a reference for the gate amplitudes.