Exact quench dynamics of the Floquet quantum East model at the deterministic point

We study the nonequilibrium dynamics of the Floquet quantum East model (a Trotterized version of the kinetically constrained quantum East spin chain) at its"deterministic point", where evolution is defined in terms of CNOT permutation gates. We solve exactly the thermalization dynamics for a broad class of initial product states by means of"space evolution". We prove: (i) the entanglement of a block of spins grows at most at one-half the maximal speed allowed by locality (i.e., half the speed of dual-unitary circuits); (ii) if the block of spins is initially prepared in a classical configuration, speed of entanglement is a quarter of the maximum; (iii) thermalization to the infinite temperature state is reached exactly in a time that scales with the size of the block.

The simplest setting for implementing kinetic constraints is in lattice systems with discrete dynamics, such as cellular automata [32,33] or quantum circuits [34].For such setups it has been possible to obtain many exact results that underpin our understanding of quantum dynamics, including on operator dynamics, information spreading, and thermalisation (see e.g.Refs.).Quantum circuits are also vital for experimental simulation of quantum systems and quantum computation, having been used to demonstrate quantum advantage, perform randomised benchmarking, and to study non-equilibrium Floquet dynamics [67][68][69][70][71][72][73][74][75][76][77].Here we consider this setting to characterise the dynamical effects of kinetic constraints by studying a circuit version of the quantum East model [78][79][80], itself a quantum generalisation of the classical East model [2].Using methods similar to those employed for dual-unitary circuits [53,61], we solve exactly the thermalization dynamics.Model setting.-Morespecifically, we consider the non-equilibrium dynamics of the Floquet Quantum East model [80] at its deterministic point, which we refer to as "deterministic Floquet quantum East" (DFQE) model.This system can be thought of as a brickwork quantum circuit, see Fig. 1, acting on a chain of 2L qubits and with local gate given by where P = 1 1 − P = (1 1 + Z)/2 is the projector to the up state |1⟩ of the qubit (the down state is denoted by |0⟩) and {X, Y, Z} are Pauli matrices [81].The quantum circuit with local gate (1) was first studied in Ref. [82] (see also Ref. [83]) and is the quantum counterpart of the classical Floquet East model of Ref. [84].The gate (1) deterministically implements the constraint that defines both the classical [2,3] and quantum [78][79][80] East models, where a site can flip only if its right neighbour is in the up state.The local gate (1) is part of the so called second hierarchy of generalised dual-unitary circuits (DU2) introduced in Ref. [85].In the jargon of quantum circuits, Eq. ( 1) is a CNOT (controlled NOT) gate [86], and as such it is a Clifford gate [87,88].This implies that there exists a class of initial stabilizer states whose dynamics can be efficiently simulated classically.Our discussion, however, is not restricted to this class.
Following a standard quantum quench protocol [89], the system is prepared in an initial state |Ψ 0 ⟩, which we take to be a product state in space, and then let to evolve unitarily as in Fig. 1b.We characterise the ensuing dy-namics using the so called space evolution approach (also known as folding algorithm) [90] (see also Refs.[91][92][93][94][95]).This can be used to characterise the evolution of general local observables [53,56,57,96], quantum information [58,60,[97][98][99][100][101][102], and even spectral properties [44, 46-48, 51, 52], but is most easily explained by considering the one-point function of an operator, O x , acting on a single qubit.We represent this via a tensor-network diagram and fold on the portions of the network representing forward and backward evolution, see Fig. 1c, and contract the network horizontally (in space rather than time).Namely, we write the one-point function using the space transfer matrices T and T O , defined in Fig. 1c, as ( Note that the transfer matrices appearing in this expressions act on the vertical folded lattice, i.e., they act on the Hilbert space H t = C 4 ⊗2t , and for simplicity we assumed the initial state to be two-site shift invariant.This latter assumption is not necessary for our analysis and will be explicitly lifted in the second part of this work.
Because of unitarity and locality of the interactions the transfer matrix T has a very simple spectrum: its only eigenvalues are 1 and 0 [57,99].The transfer matrix itself is not rank one, since the eigenvalue zero has generically a non-trivial Jordan structure.However, the size of its Jordan blocks are bounded by 2t, which implies that T 2t is rank one.It can be written as ( This expression suggests an interesting physical interpretation of the fixed points: they are the mathematical objects encoding the influence of the rest of the system on the subsystem where O acts.For this reason they are also referred-to as influence matrices [96].Equation (3) might seem to be a drastic simplification of Eq. ( 2), as it replaces a complicated matrix product with a matrix element between two fixed points.This form offers practical advantage only when the influence matrices can be computed efficiently, e.g., when they can be represented by matrix product states with low bond dimension.This is not possible in general: for generic systems and initial states, influence matrices have volume law entanglement in time [104].Some systems, however, avoid this general rule.These includes a class of chaotic dual-unitary circuits [61] evolving from a family of compatible initial states [53], and evolution from compatible states in the Rule 54 quantum cellular automaton [56,57].In fact, Ref. [105] argued that in the presence of integrability every low entanglement initial state should generate low entangled influence matrices.
Here we show that also the non-integrable DFQE admits solvable initial states generating analytically tractable influence matrices.We find three distinct families of initial states with influence matrices in dimer-product form, i.e., entangling together only pairs of sites along time.We then use this result to study the exact quench dynamics of a block of spins when the rest of the system is prepared in a solvable state.We show that, regardless of the initial state, the block relaxes to the infinite temperature state in a finite number of time steps.Moreover, we provide an exact description of the full entanglement dynamics if the block is initially prepared in a solvable state.Exact Fixed Points.-Webegin by observing that U in Eq. ( 1), see Fig. 1, obeys the local relations which define DU2 circuits [85].Relations (4) imply [85] that if the initial states of two neighbouring sites fulfil the fixed points of the transfer matrix T are of the form given in Fig. 2a.To see this consider ⟨L s | T, with ⟨L s | given in Fig. 2a.Starting from above we apply repeatedly the first of (4) until we remove the leftmost column of gates.We then proceed with removing the second column up to the gate applied on the initial state.The latter can be removed using the first of (5) while the numerical factors combine to give ⟨L s | T = ⟨L s |.Analogously, using the right relations of ( 4) and ( 5) gives Here and in the following we add the subscript "s" to quantities computed for initial states fulfilling (5).
A remarkable property of the DFQE is that we do not need to fulfil both these conditions to have simple fixed points.If |ϕ 1,2 ⟩ are both classical configurations |0⟩ or |1⟩, only the first of ( 5) is fulfilled, however the fixed points display the simple form of Fig. 2b.Similarly, if both |ϕ 1,2 ⟩ are the flat superpositions only the second of ( 5) holds but the fixed points take the simple form in Fig. 2c.This is because the local gate fulfils the following relations = , where we have introduced the diagrams and notation and ⊗ r indicates that the tensor product is between the states of the same site in the forward and backward branches, see Fig. 1a.The simplification mechanism is very similar to the one discussed after Eq. ( 5) and we refer the reader to the Supplemental Material [106].In the following we add the subscripts "cl" and "F" to quantities computed respectively for initial states that are classical configurations and flat superpositions of them [107].Subsystem Dynamics.-Theexact expressions for the fixed points in Fig. 2 represent our first main result.To illustrate their power, we use them to determine the relaxation time of a block of 2ℓ qubits in a region denoted by A. In particular, we focus on the quench from the state (5) and |Ψ ′ ⟩ is arbitrary [108].Using the exact expressions in Fig. 2a we find that the reduced density matrix at time t can be simplified as follows Note how Eq. ( 9) does not contain explicitly the environment ( Ā, the complement of A): its effect is encoded in the boundaries of the time-evolution operator of A. In other words, the superoperator formed by two subsequent horizontal layers of the tensor network in Eq. ( 9), retains information on Ā only through its depolarising boundaries.In general, tracing out part of a unitarily evolving system gives rise to non-Markovian dissipative evolution on the subsystem [109].In constrast, for the case of the DFQE the evolution of the subsystem is Markovian and the superoperator C x is a time-local quantum map.Equation ( 9) represents a drastic simplification: the dynamics of a block of 2ℓ spins can be fully determined by diagonalising a 4 2ℓ ×4 2ℓ matrix, which can be done analytically for small ℓ and numerically larger ℓ.Moreover, one can use Eq. ( 9) to show [106] that ρ A (t) = 1 1/2 2ℓ for any t ≥ 2ℓ, so that the subsystem reaches the maximal entropy state in a finite number of steps.This result contrasts with what happens in generic systems, where the presence of exponential corrections means that the stationary state is only reached exactly at infinite time.An analogous situation to the one here is found in dual-unitary circuits [49], with an important difference: in the DFQE the number of steps to approach stationarity is twice larger than for dual-unitaries.
Entanglement.-Using the above properties we can compute the growth of entanglement from various homogeneous and inhomogeneous initial states.For concreteness we consider a system that is prepared in a solvable state everywhere, except for a finite subsystem A of length ℓ = |A|.At some later time t the Rényi entanglement entropy between A and the rest is defined as where n is the Rényi index, and ρ A,Ψ ′ (t) is given in Eq. ( 9).Within A the system is prepared in one of the classes of solvable states: those fulfilling both Eqs. ( 5), homogeneous flat states, or classical configurations.The entanglement entropies show different scalings depending on the ratio between the subsystem size ℓ, and time t.In particular, we expect a simple result in the limit t → ∞, where the entropies saturate at a value extensive in ℓ (see Fig. 3b).In fact, the finite-time relaxation discussed above implies that for any initial state in A and t ≥ 2ℓ all Rényi entropies are the same For times that are shorter than 2ℓ, there are no immediate simplifications at the level of the single reduced  5) and ( 18) (red line), or one of ( 5) and one of (7) (blue line).For t > 2ℓ all curves reach the infinite-temperature value 2ℓ ln 2.
density matrix, but we need to consider the full trace in Eq. (11).Assuming for definiteness that inside of the subsystem A the initial state Ψ ′ is a product state and is invariant under lattice shifts by an even number of sites, the generalised purity can be compactly expressed in terms of the corresponding space transfer-matrix T as where P n is an operator that implements the permutation of n copies, see Fig. 3a.This expression suggests another conceptually simple regime: the "early time" regime where t is fixed and |A| is large.In this regime Eq. ( 13) is written in terms of a large power of a finite matrix expressible in terms of a fixed point: whenever t < ℓ/2 the powers of the transfer matrix factorize and ( 13) reduces to a product of two matrix elements, With the specific form of fixed points, Fig. 2, we can evaluate these overlaps and obtain for the three classes [106] Interestingly, we see that the entanglement entropies all grow with the same slope.Moreover, for classical configurations and their flat superposition this slope is reduced by 2: this is an explicit example of the initial-state dependence of the entanglement velocity.Note that for the flat superposition the range of validity of the early time expression is larger (t < ℓ rather than 2t < ℓ) due to the flat state being locally invariant under the dynamics.
The hardest regime to access is the intermediate-time regime ℓ/2 < t < 2ℓ.In this case Eq. ( 13) does not directly factorise, and cannot be determined only knowing the fixed points.Remarkably, and in contrast to other known solvable examples [56,58], Eq. ( 13) can be evaluated also in this regime for the three cases considered here.This leads to a complete description of the entanglement dynamics so far only attained for dual-unitary circuits [53,61].In particular, when the subsystem is prepared in a classical configuration, or the flat state, the partition sum (13) evaluates to 2 (1−n)t for all t in the intermediate regime, i.e., ℓ/2 < t < 2ℓ.This gives [106] Instead, if the subsystem is prepared in a solvable state that additionally satisfies the stationary state is reached at t = ℓ (rather than 2ℓ), and for ℓ/2 < t < ℓ Eq. ( 13) gives 2 2t(1−n) .This implies An interesting question is whether this surprising result can be ascribed to a general property of the initial states.While the DFQE is a Clifford circuit the above is not a consequence of initial states being stabilizer states: even though classical configurations and the flat state are stabilizers, the solvable states fulfilling Eqs. ( 5) and ( 18) are generically not.As shown in [106], the exact results in Eqs.(17) and (19) rely on three different "microscopic mechanisms" of simplifications that combine properties of the initial states and the time-evolution.The overall effect, however, is similar in the three cases: the microscopic simplifications decouple the entanglement production at the two boundaries between A and Ā.This allows to treat the problem as if it were always in the early time regime.Importantly, Eq. ( 18) is a necessary requirement for this to happen: finite-time numerics show that for solvable states that do not satisfy that condition the entanglement entropy deviates from ( 19) at intermediate times.
Conclusions.-We have solved exactly the entanglement dynamics of the deterministic Floquet quantum East model, a quantum circuit defined in terms of local CNOT gates that implement the same kinetic constraint as the East model [2,78].To our knowledge, these are the first exact results for entangling dynamics in an interacting non-integrable circuit beyond those in the dual-unitary class.The simplicity of the DFQE model allows its dynamics to be solved in the large size limit for a broad class of initial product states that extends beyond Clifford stabilizers, exploiting the techniques of propagation-in-space.One can think of many avenues for future research.An immediate one is to characterise exactly operator spreading by extending the results of Ref. [110] on the butterfly velocity of DU2 circuits to determine the full profile of out-of-time-ordered correlators.Other directions include studying the effect of local measurements on entanglement at the level of quantum trajectories [62,63,111,112] and the quantification of dynamical fluctuations as is done in the classical Floquet East [84].

Supplemental Material
Here we report some useful information complementing the main text.In particular -In Sec.we introduce the space transfer matrix and compute the corresponding fixed points for three choices of initial states.
-In Secs.we provide additional details on the computation of the entanglement entropy for three distinct time regimes.

EXACT FIXED POINTS
We define the transfer matrix in space T at time t for a given two-site state |ϕ 1 ⟩ ⊗ |ϕ 2 ⟩ as: the corresponding fixed points, i.e., left-and right-eigenvectors with largest eigenvalue λ = 1, can be computed for the three classes of states presented in the letter (results are shown in Fig. 2).In this section we want to prove they satisfy the fixed point relations for each class of states.
We first consider the states |ϕ 1 ⟩ and |ϕ 2 ⟩ fulfilling both relations of Eq. ( 5).The corresponding (normalised) fixed points are where the normalisation is chosen such that ⟨L s |R s ⟩ = 1.The left vector indeed satisfies where we start contracting the above tensor network from the top by using = 2 . (sm-4) The leftmost column is then removed by repeatedly applying the left of ( 4), and the left of ( 5) in the end.In the second last equality we further apply the left of ( 4), to contract the second column.
Similarly, the right vector |R s ⟩ satisfies: Above, the first contraction follows from unitarity and produces a factor 1/2.We then contract the rightmost column iterating the right relation in (5).The same for the other column, where we also use the right of (5) to contract the last bit, which cancels out the factor 1/2.We now consider the states |ϕ 1 ⟩ = |ϕ 2 ⟩ = |−⟩.The corresponding (normalised) fixed points are: Indeed the left vector satisfies In this case, we use the left relation in (7) to contract both the left columns from the bottom to the top.The fact that |R F ⟩ satisfies T|R F ⟩ = |R F ⟩ follows immediately from (sm-5) as the flat states fulfil the same relations used in the previous calculation.
We finally consider classical states i.e., |ϕ 1 ⟩ = |s 1 ⟩ and |ϕ 2 ⟩ = |s 2 ⟩, with s 1 , s 2 ∈ Z 2 .We have that the corresponding fixed points are: where the classical configuration {s ′ j } is given as In particular {s ′ j } is a periodic sequence with period 3. Since classical configurations fulfil the same relations as used in (sm-3), the left vector is the same as in the solvable case, ⟨L cl | = ⟨L s |.To show that |R cl ⟩ as introduced above is the right fixed point, however, it requires some additional work.We start by applying the classical transfer matrix to |R cl ⟩, without specifying s ′ j (this will be done at the end), and we contract the rightmost column by using the right of (7), where (7) implies Repeating the same steps again we obtain Simplifying this we get Imposing now the fixed point condition (i.e., s ′′′ j = s ′ j ) on (sm-13), we finally obtain the form in (sm-9).

ENTANGLEMENT ENTROPY Early-time regime
Using the explicit form of fixed-points as given in Fig. 2, we can express the factors appearing in the main-text Eq. ( 15) as a 2n × 2t tensor network can be applied also to odd numbers.In this case the 2x-th power of the transfer matrix is expressed as, where is the projector to the subspace with both legs in the same state, and the r.h.s.follows from = 2 , = . ( Now we note that the following holds, which allows us to simplify the above tensor-network into where the second equality follows from repeated application of (sm-25), together with the 2nd hierarchy condition and (sm-24).To obtain the last equality, we use unitarity together with the invariance of under .Eq. (sm-26) suggests that application of C x of at least 2x times to any initial state results in the maximum entropy state.In particular, this means that starting with the system initialized in the solvable state everywhere except possibly inside a finite subsystem A, the density matrix reduced to the subsystem A will be after 2ℓ, ℓ = |A|, exactly given by the maximum entropy state, Late-time regime for the quench from solvable states The above statement holds independently of the state inside the subsystem A. However, if the state inside the subsystem A for a subset of solvable initial states, the stationary state is reached before 2ℓ.In particular, the requirement on the initial state is that it satisfies both the second of Eq. ( 5) of the main text, and (sm-28) We start with the diagrammatic representation of the reduced density matrix at time t = ℓ, . (sm-29) Repeatedly using (sm-28) together with the manipulations used in the previous subsection, the projectors can be brought all the way to the right edge, which allows us to start applying the second hierarchy relation together with the right of (5) of main text, (sm-30) Now we note that every solution to the second relation of Eq. ( 5) in the main text satisfies also the following This gives us and therefore, in the case of this subset of initial states, the relaxation happens already at t = ℓ.

Subclass of solvable states
As the first example we treat the subclass of solvable initial states that (besides Eq. ( 5) of the main text) obeys Eq. (sm-28), in the regime where the subsystem size ℓ = |A| is smaller than 2t, but larger than t.In this case, the reduced density matrix is not yet equal to the stationary one, and at the same time the powers of the space transfer matrix do not factorize.The reduced density matrix is in this case given by the following diagram, (sm-33) To express traces of powers of the reduced density matrix we can imagine to take many copies (replicas) of the diagram above, and connect the top legs in the staggered way.By introducing dark squares as the notation for n copies of the folded gate (i.e., each leg now represents 2n qubits), and , as symbols for the two different pairings of the 2n legs we can express the trace of n-th power of ρ A (t) as (sm-37) Repeatedly applying them -together with the initial-state condition (sm-28), and the unitarity from the topallows us to rewrite (sm-35) as To progress from here, we note that the condition (sm-28) is satisfied whenever at least one of ϕ we find the following generalisation of Eq. ( 5) of the main text Here and in the following we denote by s a classical configuration on all the 2n copies.These equations in particular imply This reduces the triangle in (sm-38) into . (sm-42) The only thing left to do is to combine all the relevant factors where we remark that the second equality holds for all initial states that are both solvable and satisfy (sm-28) (i.e., they are in the subset considered here).This finally gives As a finaly point, we remark that here we never explicitly used translational invariance of the initial state, and pairs (ϕ 1 , ϕ 2 ) could be chosen independently for all pairs of sites (as long as the relevant conditions are satisfied) without changing the result.
Classical subregion in a system prepared in a solvable state Let us now consider entanglement entropy between the subsystem A and the rest, where the rest is prepared in the solvable state, while A is initialized in a classical configuration.First we treat the regime t < ℓ < 2t, and we denote the classical configuration by [s 1 , s 2 , . . ., s 2ℓ ].Diagrammatically, the trace of powers of the reduced density matrix takes the following form Using the same simplifications as in the previous section we obtain We now consider the regime t 2 < ℓ < t and compute the following . (sm-47) The r.h.s.follows by using unitarity together with the first of (sm-37), and then using the deterministic rule to evolve the initial classical state [s j ] j to the classical state [s ′ j ] j on the diagonal.We can now first remove the projectors on the left by continuously applying the r.h.s. of (sm-37), and then by combining unitarity and Eq. ( 5) the diagram contracts fully: After noting we can finally gather all the factors together and obtain Region of flat states in a system prepared in the solvable state Now we consider a quench from the system prepared in the solvable state everywhere, except for a finite region A which is initialized in the flat state, and we are interested in the entanglement entropy between A and the rest at some later time t.Since the flat state is locally invariant under time-evolution, the transfer matrix factorizes into fixed points for all times ℓ ≤ |A| (and not only for t ≤ ℓ/2).What remains to be understood is the intermediate regime with ℓ < t < 2ℓ.The trace of the n-th power of the reduced density matrix is given as where the last equality follows from the realization that is up to normalization equal to .To simplify the rest, we need to introduce a projector to the subspace of products of two-site states [1, 0, 0, 1] and [0, 1, 1, 0] in the copies connected by we obtain the r.h.s. of Eq. (sm-55).

FIG. 1 .
FIG. 1. Deterministic Floquet quantum East model.(a) Diagrammatic representation of the gate Eq. (1).Thick lines correspond to the folded representation of the forward and backward branches.(b) Time evolution of the state as a quantum circuit, |Ψ(t)⟩ = U t |Ψ0⟩.The dashed boxes indicate the initial state |Ψ0⟩ and the evolution operator U for one time step.(c) One point function of a local operator (l.h.s.) and it folded representation (r.h.s.).The dashed and full boxes outline the space transfer matrices.
where |R⟩ and |L⟩ are the right and left fixed points of T [103].In other words, for L ≥ 2t the one-point function of interest is fully specified the fixed points.In particular we have lim L→∞ ⟨Ψ(t)|O x |Ψ(t)⟩ = ⟨L|T O |R⟩ .

= 2
the r.h.s.follows by applying unitarity from the top and the local invariance of the flat state.Using now relations (sm-37), we introduce projectors on the left edge and bring them down to the diagonal.Then, using the unitarity of gates from top left, we end up with a diagonal strip tr[ρ n A (t)] = 2 n(ℓ−t) (1+n)ℓ−2nt , (sm-52)