Energy-Space Quantum Walks

• Energy-space quantum walks are quantum-walk constructions where the energy eigenstates form a ladder and transitions represent energy exchange and thermal transport.
• The framework employs classical birth–death–lazy dynamics to derive Gibbs stationary states through detailed balance and biased transport along discrete energy levels.
• A collision-assisted quantum embedding introduces a coherence parameter that decouples population relaxation from off-diagonal coherence effects, enabling thermalization without state convergence.

Energy-space quantum walks are quantum-walk constructions in which the walk space is the energy eigenbasis itself, so that the walker moves on a ladder of levels $\{\ket{n}\}_{n\ge 0}$ rather than on a position lattice. In this formulation, the projectors $M_n=\ket{n}\!\bra{n}$ define the level populations, and elementary transitions $\ket{n}\leftrightarrow\ket{n\pm1}$ represent energy exchange. The framework introduced in "Energy-space quantum walks: Thermalization without state convergence" reinterprets thermalization as transport in energy space, independently of microscopic system–bath details, and uses this perspective to study equilibration, thermalization, and irreversibility through a minimal dynamics in which population transport and coherence generation can be separated structurally (Santos et al., 14 May 2026).

1. Energy space as walk space

The defining move of an energy-space quantum walk is the mapping of the usual configuration space of a walk onto an energy ladder. The basis states are the energy eigenstates $\{\ket{n}\}_{n\ge 0}$, and the walk is organized by transitions between neighboring levels. In this setting, the fundamental observables are the level populations

$\wp_t(n)=\mathrm{Tr}[M_n\rho_t],$

with $M_n=\ket{n}\!\bra{n}$, and transport is interpreted directly as redistribution of probability along the energy spectrum (Santos et al., 14 May 2026).

This formulation is intentionally independent of a microscopic bath model. Rather than deriving equilibration from a specific system–environment Hamiltonian, it treats equilibration as an effective transport problem on a discrete ladder. In equally spaced spectra, with $E_n=nE$, the walk acquires a direct thermodynamic interpretation because stationary geometric populations become Gibbs populations. A plausible implication is that the framework isolates which aspects of thermalization depend only on ladder transport and which require genuinely microscopic assumptions.

The energy-space viewpoint also sharpens the distinction between population relaxation and full-state relaxation. In the minimal model, the energy basis is not merely a measurement basis but the dynamical substrate on which irreversibility-like behavior is formulated. This is why the central claim is not only equilibration, but thermalization without state convergence: the energy distribution can relax while the density operator remains nonthermal (Santos et al., 14 May 2026).

2. Classical birth–death–lazy dynamics and Gibbs stationary states

The classical limit of the framework is a stochastic walk on the nonnegative integers, specifically a birth–death–lazy dynamics. The map $\Phi$ acts on the basis projectors as

$$\Phi(M_n)= \begin{cases} (p_-+p_0) M_0 + p_+ M_1, & n=0,\ p_- M_{n-1} + p_0 M_n + p_+ M_{n+1}, & n\ge 1, \end{cases}$$

with

$p_+,p_-,p_0\ge 0,\qquad p_++p_-+p_0=1.$

Accordingly, the populations obey

$M_n=\ket{n}\!\bra{n}$0

and, for $M_n=\ket{n}\!\bra{n}$1,

$M_n=\ket{n}\!\bra{n}$2

The three parameters have the direct interpretation: $M_n=\ket{n}\!\bra{n}$3 is an upward step in energy, $M_n=\ket{n}\!\bra{n}$4 a downward step, and $M_n=\ket{n}\!\bra{n}$5 a lazy or stay-put step (Santos et al., 14 May 2026).

Under the biased condition $M_n=\ket{n}\!\bra{n}$6, the chain has a unique normalizable stationary distribution. Introducing the stationary current

$M_n=\ket{n}\!\bra{n}$7

stationarity implies $M_n=\ket{n}\!\bra{n}$8, and the boundary condition $M_n=\ket{n}\!\bra{n}$9 forces vanishing current throughout the ladder. The resulting local balance condition,

$\ket{n}\leftrightarrow\ket{n\pm1}$0

gives

$\ket{n}\leftrightarrow\ket{n\pm1}$1

The stationary state is therefore geometric (Santos et al., 14 May 2026).

For equally spaced energies $\ket{n}\leftrightarrow\ket{n\pm1}$2, the same stationary law is Gibbs:

$\ket{n}\leftrightarrow\ket{n\pm1}$3

or equivalently

$\ket{n}\leftrightarrow\ket{n\pm1}$4

The model thus shows that Gibbs statistics can emerge from biased transport in energy space, even without an explicit thermal bath (Santos et al., 14 May 2026).

The framework also admits level-dependent rates $\ket{n}\leftrightarrow\ket{n\pm1}$5. In that case,

$\ket{n}\leftrightarrow\ket{n\pm1}$6

for $\ket{n}\leftrightarrow\ket{n\pm1}$7, with a suitable boundary action at $\ket{n}\leftrightarrow\ket{n\pm1}$8. If the stationary current vanishes, then

$\ket{n}\leftrightarrow\ket{n\pm1}$9

A sufficient condition for a Gibbs stationary state is the local detailed-balance relation

$\{\ket{n}\}_{n\ge 0}$0

In this generalization, the lazy term affects relaxation times but not the stationary ratios (Santos et al., 14 May 2026).

3. Collision-assisted quantum embedding

The quantum extension is a unitary, collision-assisted embedding of the classical walk. The system interacts sequentially with a three-level ancilla whose states

$\{\ket{n}\}_{n\ge 0}$1

encode the upward, lazy, and downward channels. The shift structure is built from the ladder operators

$\{\ket{n}\}_{n\ge 0}$2

The unitary step is

$\{\ket{n}\}_{n\ge 0}$3

with $\{\ket{n}\}_{n\ge 0}$4 and $\{\ket{n}\}_{n\ge 0}$5 (Santos et al., 14 May 2026).

The ancilla is prepared in the one-parameter family

$\{\ket{n}\}_{n\ge 0}$6

where

$\{\ket{n}\}_{n\ge 0}$7

and

$\{\ket{n}\}_{n\ge 0}$8

The parameter $\{\ket{n}\}_{n\ge 0}$9 is the single coherence control parameter: $\wp_t(n)=\mathrm{Tr}[M_n\rho_t],$0 is the incoherent classical limit, while $\wp_t(n)=\mathrm{Tr}[M_n\rho_t],$1 introduces coherent quantum corrections (Santos et al., 14 May 2026).

The reduced system dynamics is

$\wp_t(n)=\mathrm{Tr}[M_n\rho_t],$2

For $\wp_t(n)=\mathrm{Tr}[M_n\rho_t],$3,

$\wp_t(n)=\mathrm{Tr}[M_n\rho_t],$4

which reproduces the classical birth–death–lazy process on diagonal states. For general $\wp_t(n)=\mathrm{Tr}[M_n\rho_t],$5, the explicit reduced map is

$\wp_t(n)=\mathrm{Tr}[M_n\rho_t],$6

This formula is the basic quantum dynamical law of the model (Santos et al., 14 May 2026).

A major structural result is that population dynamics decouples from coherence generation. The term

$\wp_t(n)=\mathrm{Tr}[M_n\rho_t],$7

is purely off-diagonal in the energy basis and therefore does not affect the diagonal populations. Consequently, the populations obey

$\wp_t(n)=\mathrm{Tr}[M_n\rho_t],$8

exactly as in the classical process, independently of $\wp_t(n)=\mathrm{Tr}[M_n\rho_t],$9. The diagonal sector evolves autonomously as a classical Markov chain, while the off-diagonal sector is driven by coherence injected at the boundary; there is no back-action from coherence onto populations (Santos et al., 14 May 2026).

4. Thermal distance and thermalization without state convergence

To distinguish equilibration from thermalization, the framework uses the trace distance

$M_n=\ket{n}\!\bra{n}$0

At each time $M_n=\ket{n}\!\bra{n}$1, one introduces a Gibbs state $M_n=\ket{n}\!\bra{n}$2 having the same instantaneous mean energy as $M_n=\ket{n}\!\bra{n}$3,

$M_n=\ket{n}\!\bra{n}$4

and defines the thermal distance

$M_n=\ket{n}\!\bra{n}$5

For diagonal states, this reduces to total variation distance on the populations:

$M_n=\ket{n}\!\bra{n}$6

For equally spaced levels $M_n=\ket{n}\!\bra{n}$7, the matching Gibbs distribution is geometric,

$M_n=\ket{n}\!\bra{n}$8

with

$M_n=\ket{n}\!\bra{n}$9

This diagnostic measures distance to the Gibbs manifold rather than merely to the eventual stationary state (Santos et al., 14 May 2026).

The central claim of the model is that what converges and what fails to converge are different sectors of the dynamics. If the classical process is ergodic and satisfies the balance conditions, then the populations converge to the stationary distribution, and under detailed balance this stationary distribution is Gibbs. At the level of energy populations, one therefore has thermalization. But for $E_n=nE$0, the full density matrix does not converge to the Gibbs state, because the dynamics generates persistent off-diagonal coherence (Santos et al., 14 May 2026).

This failure is visible already at first order. Even if the initial state is Gibbs,

$E_n=nE$1

so $E_n=nE$2 is not a fixed point for any $E_n=nE$3. Hence the model realizes thermalization without state convergence: populations become Gibbsian, but the density operator approaches a different stationary state with coherence (Santos et al., 14 May 2026).

The perturbative analysis makes the same point quantitatively. Writing

$E_n=nE$4

one has

$E_n=nE$5

Defining the classical thermal distance

$E_n=nE$6

the perturbative bound is

$E_n=nE$7

The boundary occupation sum satisfies

$E_n=nE$8

and the exact dynamics settles to a finite asymptotic thermal distance

$E_n=nE$9

The long-time deviation from Gibbs is therefore controlled by the coherence parameter and linked perturbatively to classical transport properties, especially the boundary occupation (Santos et al., 14 May 2026).

5. Momentum-space implementations as experimental templates

Although the 2026 formulation is abstract, closely related non-position-space walks have already been realized experimentally in momentum space. In the first experimental realization of a quantum walk in momentum space, a Bose–Einstein condensate of ultracold $\Phi$0Rb atoms was used with two hyperfine states,

$\Phi$1

as the coin space and the center-of-mass momentum states $\Phi$2 as the walk space. The discrete-time protocol had the standard form

$\Phi$3

with a microwave coin and a conditional momentum shift implemented through a quantum resonant ratchet based on the atom-optics kicked rotor. The experiment demonstrated ballistic spreading, a two-peak structure, a quantum-to-classical transition under coin-phase noise, steering via biased coin and biased ratchet settings, and reversal by applying the conjugate step operator (Dadras et al., 2018).

The subsequent comprehensive study reported an experimental discrete-time quantum walk in momentum space using a spinor Bose–Einstein condensate of ultra-cold $\Phi$4Rb atoms, with about 70,000 atoms initially in $\Phi$5. The walk space was the discrete momentum ladder

$\Phi$6

the shift was generated by a quantum resonant ratchet at the Talbot-time condition $\Phi$7, and the measured observable was

$\Phi$8

The experiment showed two ballistically separating peaks and a standard deviation growing approximately linearly with the number of steps. It also quantified the effects of shift strength, phase compensation, noise, quasimomentum width, and initialization, and reported that the mean energy increases with step number, reflecting the ballistic spreading (Dadras et al., 2018).

These momentum-space walks are directly relevant to energy-space quantum walks because the walker does not move on a spatial lattice: it moves through momentum classes tied to kinetic energy. In the experimental discussion, momentum classes were described as directly tied to kinetic energy, and the observables included mean energy growth. This suggests that the momentum-space platform is a concrete experimentally realized template for broader walks in momentum or energy space, where transport occurs through energy-changing kicks rather than through literal spatial displacement (Dadras et al., 2018).

6. Related non-position-space formulations and conceptual extensions

Energy-space quantum walks belong to a broader family of quantum walks defined on noncanonical walk spaces. One explicit example is the coined discrete-time quantum walk on a finite toral phase space, where one coin branch translates in position and the other in conjugate momentum:

$\Phi$9

In this model, the phase space is a discrete torus, the spectrum is exactly solvable, odd-$$\Phi(M_n)= \begin{cases} (p_-+p_0) M_0 + p_+ M_1, & n=0,\ p_- M_{n-1} + p_0 M_n + p_+ M_{n+1}, & n\ge 1, \end{cases}$$0 lattices have equally spaced eigenangles independent of the coin angle, and the dynamics shows ballistic spreading, exact periodicity, cat-state formation in phase space, participation-ratio growth $$\Phi(M_n)= \begin{cases} (p_-+p_0) M_0 + p_+ M_1, & n=0,\ p_- M_{n-1} + p_0 M_n + p_+ M_{n+1}, & n\ge 1, \end{cases}$$1 with $$\Phi(M_n)= \begin{cases} (p_-+p_0) M_0 + p_+ M_1, & n=0,\ p_- M_{n-1} + p_0 M_n + p_+ M_{n+1}, & n\ge 1, \end{cases}$$2 over an intermediate regime, and an Ehrenfest time scaling $$\Phi(M_n)= \begin{cases} (p_-+p_0) M_0 + p_+ M_1, & n=0,\ p_- M_{n-1} + p_0 M_n + p_+ M_{n+1}, & n\ge 1, \end{cases}$$3 (Omanakuttan et al., 2018).

Another adjacent formulation arises in trapped-ion phase-space walks with non-orthogonal position states. There the natural “positions” are coherent states $$\Phi(M_n)= \begin{cases} (p_-+p_0) M_0 + p_+ M_1, & n=0,\ p_- M_{n-1} + p_0 M_n + p_+ M_{n+1}, & n\ge 1, \end{cases}$$4 with overlaps

$$\Phi(M_n)= \begin{cases} (p_-+p_0) M_0 + p_+ M_1, & n=0,\ p_- M_{n-1} + p_0 M_n + p_+ M_{n+1}, & n\ge 1, \end{cases}$$5

and the walk can be mapped exactly to an ordinary discrete-time quantum walk on an orthonormal lattice by introducing the Gram operator

$$\Phi(M_n)= \begin{cases} (p_-+p_0) M_0 + p_+ M_1, & n=0,\ p_- M_{n-1} + p_0 M_n + p_+ M_{n+1}, & n\ge 1, \end{cases}$$6

the dual basis $$\Phi(M_n)= \begin{cases} (p_-+p_0) M_0 + p_+ M_1, & n=0,\ p_- M_{n-1} + p_0 M_n + p_+ M_{n+1}, & n\ge 1, \end{cases}$$7, and the orthonormal basis

$$\Phi(M_n)= \begin{cases} (p_-+p_0) M_0 + p_+ M_1, & n=0,\ p_- M_{n-1} + p_0 M_n + p_+ M_{n+1}, & n\ge 1, \end{cases}$$8

In that representation, non-orthogonality is transferred entirely into an extended initial state and a momentum-space weight $$\Phi(M_n)= \begin{cases} (p_-+p_0) M_0 + p_+ M_1, & n=0,\ p_- M_{n-1} + p_0 M_n + p_+ M_{n+1}, & n\ge 1, \end{cases}$$9, which determines which group velocities are sampled. The same formalism permits momentum shifts

$p_+,p_-,p_0\ge 0,\qquad p_++p_-+p_0=1.$0

and stepwise drift $p_+,p_-,p_0\ge 0,\qquad p_++p_-+p_0=1.$1, producing an analog of Bloch oscillations (Matjeschk et al., 2012).

A different but related mechanism appears in electric quantum walks in two dimensions. There the field is introduced by a phase factor

$p_+,p_-,p_0\ge 0,\qquad p_++p_-+p_0=1.$2

which in momentum space shifts $p_+,p_-,p_0\ge 0,\qquad p_++p_-+p_0=1.$3 at each step. The resulting transport is governed by the geometry of the quasi-energy bands, especially by the presence or absence of conical intersections. In Grover and DFT walks, the field produces direction-dependent transient trapping, while in the alternate walk with $p_+,p_-,p_0\ge 0,\qquad p_++p_-+p_0=1.$4 and in the 2D Hadamard walk it can produce perfect 2D trapping. This suggests a broader quasi-energy-space interpretation in which the dynamics is controlled by repeated motion through dispersion surfaces rather than solely by transport in position space (Bru et al., 2015).

Energy-space language also appears in continuous-time settings. The ordered-Hamming-scheme construction shows that hopping of a single excitation on triangular spin lattices with non-uniform couplings and local magnetic fields can be understood as projections of quantum walks on graphs of the ordered Hamming scheme of depth 2. In that setting, the walk is interpreted as a single excitation moving in energy space of the spin network, the spectral data are governed by Tratnik bivariate Krawtchouk polynomials, and the model exhibits perfect state transfer and fractional revival for specific parameter relations (Miki et al., 2017).

Taken together, these works indicate that energy-space quantum walks are part of a larger program in which quantum walks are formulated on momentum ladders, phase-space tori, non-orthogonal coherent-state lattices, graph-reduced excitation spaces, and geometry-dependent simplicial or curved-spacetime structures. A plausible implication is that the energy-space formulation is distinguished not by abandoning the walk paradigm, but by relocating the walk space from position to a spectrally meaningful basis in which equilibration, transport, and coherence can be analyzed with unusual clarity (Arrighi et al., 2016, Nzongani et al., 2024).