Algebraic OTOC: Polynomial Growth in Quantum Walks

• Algebraic OTOC is characterized by polynomial (t²) growth in operator non-commutativity, indicating slow, non-chaotic dynamics.
• In coined quantum walks, different subspace interactions (coin–coin, walker–walker, coin–walker) yield distinct growth and saturation behaviors.
• Large-N approximations reveal quadratic scaling laws that clearly distinguish algebraic spreading from the exponential sensitivity seen in chaotic systems.

Algebraic Out-of-Time-Order Correlator (A-OTOC) denotes an out-of-time-ordered correlator whose growth is polynomial rather than exponential in time. In the coined discrete quantum walk studied by Omanakuttan and Lakshminarayan, the OTOC provides a diagnostic of operator non-commutativity across coin and walker subspaces, and all three operator choices considered—coin–coin, walker–walker, and coin–walker—display either saturation or algebraic growth rather than an early exponential instability (Omanakuttan et al., 2019). In this setting, the label “A-OTOC” is used for such polynomial behavior, with the quadratic law $F(t)\propto t^2$ serving as the characteristic signature of slow operator spreading and the absence of intrinsic quantum chaos.

1. Definition and diagnostic role

The standard OTOC is defined for Heisenberg-evolved operators $W(t)=e^{iHt}We^{-iHt}$ and $V(0)=V$ on a Hilbert space of total dimension $D$ by

$F(t)=\frac{1}{2D}\,\bigl\|[W(t),V(0)]\bigr\|_F^2 =1-\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr].$

Equivalently,

$C_4(t)=\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr], \qquad C(t)=-\,\langle [W(t),V]^2\rangle,$

so that $F(t)=1-C_4(t)$ (Omanakuttan et al., 2019).

Within this framework, $F(t)$ measures the extent to which an initially chosen perturbation fails to commute with another operator at a later time. The conventional contrast is between chaotic many-body systems, where one often finds an early exponential rise $F(t)\sim e^{\lambda_L t}$ with $\lambda_L$ the Lyapunov exponent, and non-chaotic or integrable systems, where the growth is typically power-law. The coined quantum walk realizes the latter case. The terminology “Algebraic OTOC” or A-OTOC refers precisely to this polynomial regime.

A frequent misunderstanding is to equate any increase in $W(t)=e^{iHt}We^{-iHt}$0 with quantum chaos. The coined-walk results show that this is too coarse a criterion: a nonzero and even systematic increase of the OTOC can occur in a system whose dynamics remain non-chaotic, provided the increase is algebraic rather than exponential.

2. Coined discrete quantum walk as the underlying dynamical system

The relevant Hilbert space is

$W(t)=e^{iHt}We^{-iHt}$1

with $W(t)=e^{iHt}We^{-iHt}$2, spanned by $W(t)=e^{iHt}We^{-iHt}$3, and $W(t)=e^{iHt}We^{-iHt}$4, corresponding to sites $W(t)=e^{iHt}We^{-iHt}$5 on a ring (Omanakuttan et al., 2019).

The walker sector carries the position shift

$W(t)=e^{iHt}We^{-iHt}$6

and the momentum shift

$W(t)=e^{iHt}We^{-iHt}$7

with $W(t)=e^{iHt}We^{-iHt}$8 and $W(t)=e^{iHt}We^{-iHt}$9. The coin toss is a one-parameter family

$V(0)=V$0

A single time step is generated by

$V(0)=V$1

where, in the walker-momentum basis,

$V(0)=V$2

This block-diagonalization into $V(0)=V$3 momentum-sector matrices is structurally central. It permits explicit evaluation of the OTOC for operator choices localized in different subspaces and makes it possible to derive asymptotic algebraic laws by diagonalizing each block and approximating sums by integrals. A plausible implication is that the transparency of the coined-walk spectrum is one reason the OTOC fails to develop the exponential sensitivity associated with strongly scrambling dynamics.

3. Coin–coin A-OTOC

For the coin–coin case, the perturbing operators are chosen as

$V(0)=V$4

This probes the growth of non-commutativity entirely within the coin sector while the walker acts as an auxiliary degree of freedom (Omanakuttan et al., 2019).

The exact marginal cases $V(0)=V$5 and $V(0)=V$6 yield rapid oscillations followed by saturation at $V(0)=V$7 or zero every other step. For the Hadamard walk $V(0)=V$8, the asymptotic form is

$V(0)=V$9

where $D$0 is the Bessel function. At large $D$1, the OTOC oscillates about

$D$2

with an algebraic envelope $D$3.

This behavior differs qualitatively from the $D$4 growth found in the mixed and walker-localized cases. The characteristic time scale is $D$5, so the coin–coin OTOC reaches its asymptotic oscillatory regime essentially immediately compared with the $D$6-dependent scales of the other sectors. This suggests that perturbations confined to the two-dimensional coin subspace have only a limited capacity for sustained spreading.

4. Walker–walker A-OTOC

For the walker–walker case, both operators act in the walker momentum sector: $D$7 The corresponding four-point function can be written as

$D$8

where

$D$9

and $F(t)=\frac{1}{2D}\,\bigl\|[W(t),V(0)]\bigr\|_F^2 =1-\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr].$0 under the smoothness approximation (Omanakuttan et al., 2019).

For the Hadamard choice $F(t)=\frac{1}{2D}\,\bigl\|[W(t),V(0)]\bigr\|_F^2 =1-\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr].$1, expansion of $F(t)=\frac{1}{2D}\,\bigl\|[W(t),V(0)]\bigr\|_F^2 =1-\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr].$2 and replacement of the $F(t)=\frac{1}{2D}\,\bigl\|[W(t),V(0)]\bigr\|_F^2 =1-\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr].$3-sum by an integral give

$F(t)=\frac{1}{2D}\,\bigl\|[W(t),V(0)]\bigr\|_F^2 =1-\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr].$4

More generally,

$F(t)=\frac{1}{2D}\,\bigl\|[W(t),V(0)]\bigr\|_F^2 =1-\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr].$5

The salient feature is the coexistence of quadratic growth and a long $F(t)=\frac{1}{2D}\,\bigl\|[W(t),V(0)]\bigr\|_F^2 =1-\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr].$6 time window. Among the three cases, this is the slowest-growing OTOC in system-size units, since the prefactor scales as $F(t)=\frac{1}{2D}\,\bigl\|[W(t),V(0)]\bigr\|_F^2 =1-\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr].$7. Physically, the result indicates that operator spreading within the walker sector is parametrically suppressed in the large-$F(t)=\frac{1}{2D}\,\bigl\|[W(t),V(0)]\bigr\|_F^2 =1-\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr].$8 limit, even though the growth law remains quadratic.

5. Coin–walker A-OTOC

The mixed case couples the two subspaces directly: $F(t)=\frac{1}{2D}\,\bigl\|[W(t),V(0)]\bigr\|_F^2 =1-\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr].$9 This choice measures how a coin-localized perturbation fails to commute at later times with a walker-localized one (Omanakuttan et al., 2019).

Using the same derivative and continuum approximations, the Hadamard walk satisfies

$C_4(t)=\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr], \qquad C(t)=-\,\langle [W(t),V]^2\rangle,$0

A compact fit is

$C_4(t)=\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr], \qquad C(t)=-\,\langle [W(t),V]^2\rangle,$1

valid up to $C_4(t)=\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr], \qquad C(t)=-\,\langle [W(t),V]^2\rangle,$2.

Relative to the walker–walker case, the mixed OTOC grows on the shorter scale $C_4(t)=\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr], \qquad C(t)=-\,\langle [W(t),V]^2\rangle,$3 and carries the larger finite-size prefactor $C_4(t)=\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr], \qquad C(t)=-\,\langle [W(t),V]^2\rangle,$4 rather than $C_4(t)=\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr], \qquad C(t)=-\,\langle [W(t),V]^2\rangle,$5. This places coin–walker scrambling in an intermediate regime: it is still algebraic and therefore non-chaotic, but it is less suppressed than purely walker-sector spreading. A plausible implication is that coupling across subspaces accelerates operator growth without inducing exponential sensitivity.

6. Approximations, time scales, and interpretation as absence of quantum chaos

The asymptotic A-OTOC formulas rely on two large-$C_4(t)=\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr], \qquad C(t)=-\,\langle [W(t),V]^2\rangle,$6 approximations. First, second differences such as $C_4(t)=\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr], \qquad C(t)=-\,\langle [W(t),V]^2\rangle,$7 are replaced by $C_4(t)=\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr], \qquad C(t)=-\,\langle [W(t),V]^2\rangle,$8, which is valid when $C_4(t)=\frac{1}{D}\,\Re\!\bigl[\Tr\bigl(W(t)\,V\,W(t)\,V\bigr)\bigr], \qquad C(t)=-\,\langle [W(t),V]^2\rangle,$9 and $F(t)=1-C_4(t)$0 varies smoothly with $F(t)=1-C_4(t)$1. Second, sums $F(t)=1-C_4(t)$2 are replaced by integrals, again assuming large $F(t)=1-C_4(t)$3 (Omanakuttan et al., 2019).

The resulting characteristic time scales separate the three operator configurations:

• Coin–coin: $F(t)=1-C_4(t)$4
• Coin–walker: $F(t)=1-C_4(t)$5
• Walker–walker: $F(t)=1-C_4(t)$6

Across all cases, the central conclusion is that the coined discrete quantum walk exhibits either rapid saturation or polynomial OTOC growth, not the early exponential increase associated with strong scrambling. The quadratic rise $F(t)=1-C_4(t)$7 is therefore interpreted as signaling slow “ballistic” operator growth without exponential sensitivity. In the terminology of the exposition, these are Algebraic OTOCs.

The broader conceptual proposal is to regard the coined quantum walk as a benchmark for slow operator spreading with subspace-dependent algebraic laws. The exposition further notes that similar algebraic OTOCs have been observed in integrable spin chains, many-body localized phases, and free-fermion models, and suggests a possible classification by the power-law exponent $F(t)=1-C_4(t)$8 in $F(t)=1-C_4(t)$9, the subspace dimensions, and the nature of dynamical constraints. It also identifies future directions including higher-dimensional coins or graphs, quantum walks with classically chaotic coins, and many-body generalizations in which walker degrees of freedom interact. These latter points indicate a programmatic extension of the A-OTOC concept beyond the simplest coined walk, while the explicit results of the model already establish its core meaning: polynomial operator growth as a diagnostic distinct from quantum chaos.