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# A DUQC which generate a square lattice cluster state mapped to one dimension. The DUQC in Eq.(45), (46), (47), and (48) can be written as (a), and it is equal to a quantum circuit (b), which include quantum gates acting on two distant qubits. The final state of (b) is equivalent to a square lattice cluster state (c) by deviding a 1D qubit array into 2m and rearranging it to a square lattice.

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Quantum circuits that are classically simulatable tell us when quantum computation becomes less powerful than or equivalent to classical computation. Such classically simulatable circuits are of importance because they illustrate what makes universal quantum computation different from classical computers. In this work, we propose a novel family of...

## Contexts in source publication

**Context 1**

... U cluster (t) at other odd and even times are SWAP 1 and SWAP 2 , respectively. Graphically, the above DUQC can be written as Fig. 3 (a), and it is equivalent to Fig. 3 (b). As shown in Fig. 3 (b) and (c), by rearranging the 1D qubit array (b) to the square lattice (c), the final state of the DUQC is equivalent to the square lattice cluster state. Moreover, because dual-unitary gates include arbitrary single-qubit gates, we can measure the final state in an arbitrary ...

**Context 2**

... U cluster (t) at other odd and even times are SWAP 1 and SWAP 2 , respectively. Graphically, the above DUQC can be written as Fig. 3 (a), and it is equivalent to Fig. 3 (b). As shown in Fig. 3 (b) and (c), by rearranging the 1D qubit array (b) to the square lattice (c), the final state of the DUQC is equivalent to the square lattice cluster state. Moreover, because dual-unitary gates include arbitrary single-qubit gates, we can measure the final state in an arbitrary measurement basis. Therefore, sampling ...

**Context 3**

... U cluster (t) at other odd and even times are SWAP 1 and SWAP 2 , respectively. Graphically, the above DUQC can be written as Fig. 3 (a), and it is equivalent to Fig. 3 (b). As shown in Fig. 3 (b) and (c), by rearranging the 1D qubit array (b) to the square lattice (c), the final state of the DUQC is equivalent to the square lattice cluster state. Moreover, because dual-unitary gates include arbitrary single-qubit gates, we can measure the final state in an arbitrary measurement basis. Therefore, sampling from depth-(N − √ 2N /2 ...

**Context 4**

... t) is unlikely to be classically simulatable because of the following reason. C 2 (r, t) can be graphically represented as Fig. 5 (b). After contracting unitary and dual-unitary gates of Fig. 5 (b) by using Eq. (9) and Eq. (11), the remaining gates of it can be represented as Fig. 5 (c). This is similar to correlation functions of 1D DUQCs (see Fig. 3 of Ref. [24]), but important difference is that uncontracted gates form a 2D tensor-network in 2D DUQCs. Because of this, calculating correlation functions in 2D DUQCs seems to be hard for a classical computer. Therefore, classical simulatability of correlation functions seems to depend on the relative position of two local ...

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## Citations

... Bipartite unitary gates that remain unitary under realignment are maximally entangled unitaries and thus are highly non-local [14]. Maximally entangled unitaries are called dual unitary gates in quantum many-body physics and been studied quite extensively in the recent past [24,25,26,27,28,29,30,31,32,33,34,35,36]. In (1+1)-dimensional discrete space time, dual unitary gates remain unitary when the temporal and spatial axes are exchanged, and this property plays a pivotal role in obtaining exact analytical results in dual unitary quantum circuits [37,38,39]. ...

Dual unitary gates are highly non-local two-qudit unitary gates that have been studied extensively in quantum many-body physics and quantum information in the recent past. A special subset of dual unitary gates consists of rank-four perfect tensors, which are equivalent to highly entangled multipartite pure states called absolutely maximally entangled (AME) states. In this work, numerical and analytical constructions of dual unitary gates and perfect tensors that are diagonal in a special maximally entangled basis are presented. The main ingredient in our construction is a phase-valued (unimodular) two-dimensional array whose discrete Fourier transform is also unimodular. We obtain perfect tensors for several local Hilbert space dimensions, particularly, in dimension six. A perfect tensor in local dimension six is equivalent to an AME state of four qudits, denoted as AME(4,6), and such a state cannot be constructed from existing constructions of AME states based on error-correcting codes and graph states. The existence of AME(4,6) states featured in well-known open problem lists in quantum information, and was settled positively in Phys. Rev. Lett. 128 080507 (2022). We provide an explicit construction of perfect tensors in local dimension six that can be written in terms of controlled unitary gates in the computational basis, making them amenable for quantum circuit implementations.

... These are a class of quantum circuits constructed as a brickwork pattern of two-qudit gates, which are unitary in both temporal and spatial directions. This special property enables the exact calculation of some system properties that would ordinarily be prohibitively hard to calculate [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. ...

Dual-unitary circuits are a class of quantum systems for which exact calculations of various quantities are possible, even for circuits that are non-integrable. The array of known exact results paints a compelling picture of dual-unitary circuits as rapidly thermalising systems. However, in this work, we present a method to construct dual-unitary circuits for which some simple initial states fail to thermalise, despite the circuits being "maximally chaotic", ergodic and mixing. This is achieved by embedding quantum many-body scars in a circuit of arbitrary size and local Hilbert space dimension. We support our analytic results with numerical simulations showing the stark contrast in the rate of entanglement growth from an initial scar state compared to non-scar initial states. Our results are well suited to an experimental test, due to the compatibility of the circuit layout with the native structure of current digital quantum simulators.

... Dual unitarity was shown to enable analytical com-putations of correlation functions [4,5], chaos indicator spectral form factor [6,7], operator and entanglement spreading [5,[8][9][10][11][12][13][14], deep thermalization through emergent state designs [15][16][17], study of eigenstate thermalization [18] and temporal entanglement [19][20][21]. They also proved useful in connections with measurement induced phase transitions [22][23][24], had aspects of their computational power characterized [25], and have already been realized in experimental setups [26,27]. The exhaustive parametrization is simple and is known for dual-unitary gates of two qubits [4]. ...

Quantum dynamics with local interactions in lattice models display rich physics, but is notoriously hard to study. Dual-unitary circuits allow for exact answers to interesting physical questions in clean or disordered one- and higher-dimensional quantum systems. However, this family of models shows some non-universal features, like vanishing correlations inside the light-cone and instantaneous thermalization of local observables. In this work we propose a generalization of dual-unitary circuits where the exactly calculable spatial-temporal correlation functions display richer behavior, and have non-trivial thermalization of local observables. This is achieved by generalizing the single-gate condition to a hierarchy of multi-gate conditions, where the first level recovers dual-unitary models, and the second level exhibits these new interesting features. We also extend the discussion and provide exact solutions to correlators with few-site observables and discuss higher-orders, including the ones after a quantum quench. In addition, we provide exhaustive parametrizations for qubit cases, and propose a new family of models for local dimensions larger than two, which also provides a new family of dual-unitary models.

... Specifically, here we consider a particular class of local quantum circuits known as "dual unitary circuits" [57], which are defined by the property that their bulk dynamics remain unitary also when exchanging the roles of space and time. The most remarkable feature of these systems is that, despite being quantum chaotic, they allow for exact calculations of many relevant many-body quantities [27,50,[58][59][60][61][62][63][64][65][66][67][68][69]. Surprisingly, even the very quantum chaotic nature of dual-unitary circuits can be rigorously proven [45,49]. ...

Dual-unitary circuits are a class of locally interacting quantum many-body systems displaying unitary dynamics also when the roles of space and time are exchanged. These systems have recently emerged as a remarkable framework where certain features of many-body quantum chaos can be studied exactly. In particular, they admit a class of “solvable” initial states for which, in the thermodynamic limit, one can access the full nonequilibrium dynamics. This reveals a surprising property: when a dual-unitary circuit is prepared in a solvable state the quantum entanglement between two complementary spatial regions grows at the maximal speed allowed by the local structure of the evolution. Here we investigate the fate of this property when the system is prepared in a generic pair-product state. We show that in this case, the entanglement increment during a time step is submaximal for finite times, however, it approaches the maximal value in the infinite-time limit. This statement is proven rigorously for dual-unitary circuits generating high enough entanglement, while it is argued to hold for the entire class.

... Such complexity phase transitions have initially been discussed when studying the hardness regime of the random k-SAT problem [28,29]. More recently, the complexity of classically simulating quantum circuits has been found to undergo a transition for instantaneous quantum polynomial (IQP) circuits [30][31][32], linear optical circuits [33][34][35][36], random quantum circuits [37,38], and dual-unitary circuits [39][40][41][42][43]. Such transitions are of particular interest for drawing and delineating the boundaries between the power of classical and quantum computing [44]. ...

Recently, the dynamics of quantum systems that involve both unitary evolution and quantum measurements have attracted attention due to the exotic phenomenon of measurement-induced phase transitions. The latter refers to a sudden change in a property of a state of $n$ qubits, such as its entanglement entropy, depending on the rate at which individual qubits are measured. At the same time, quantum complexity emerged as a key quantity for the identification of complex behaviour in quantum many-body dynamics. In this work, we investigate the dynamics of the quantum state complexity in monitored random circuits, where $n$ qubits evolve according to a random unitary circuit and are individually measured with a fixed probability at each time step. We find that the evolution of the exact quantum state complexity undergoes a phase transition when changing the measurement rate. Below a critical measurement rate, the complexity grows at least linearly in time until saturating to a value $e^{\Omega(n)}$. Above, the complexity does not exceed $\operatorname{poly}(n)$. In our proof, we make use of percolation theory to find paths along which an exponentially long quantum computation can be run below the critical rate, and to identify events where the state complexity is reset to zero above the critical rate. We lower bound the exact state complexity in the former regime using recently developed techniques from algebraic geometry. Our results combine quantum complexity growth, phase transitions, and computation with measurements to help understand the behavior of monitored random circuits and to make progress towards determining the computational power of measurements in many-body systems.

... Dual-unitary gates have also been used to prove deep thermalization through emergent state designs [29,30,31], study eigenstate thermalization [32], temporal entanglement [33], and aspects of their computational power have been characterised [34]. They were also generalised to the case of having three directions [35,36], as well as utilised in connections with measurement induced phase transitions [37,38,39]. ...

Exact solutions in interacting many-body systems are scarce but extremely valuable since they provide insights into the dynamics. Dual-unitary models are examples in one spatial dimension where this is possible. These brick-wall quantum circuits consist of local gates, which remain unitary not only in time, but also when interpreted as evolutions along the spatial directions. However, this setting of unitary dynamics does not directly apply to real-world systems due to their imperfect isolation, and it is thus imperative to consider the impact of noise to dual-unitary dynamics and its exact solvability. In this work we generalise the ideas of dual-unitarity to obtain exact solutions in noisy quantum circuits, where each unitary gate is substituted by a local quantum channel. Exact solutions are obtained by demanding that the noisy gates yield a valid quantum channel not only in time, but also when interpreted as evolutions along one or both of the spatial directions and possibly backwards in time. This gives rise to new families of models that satisfy different combinations of unitality constraints along the space and time directions. We provide exact solutions for the spatio-temporal correlation functions, spatial correlations after a quantum quench, and the structure of steady states for these families of models. We show that noise unbiased around the dual-unitary family leads to exactly solvable models, even if dual-unitarity is strongly violated. We prove that any channel unital in both space and time directions can be written as an affine combination of a particular class of dual-unitary gates. Finally, we extend the definition of solvable initial states to matrix-product density operators. We completely classify them when their tensor admits a local purification.

... From here we get that invariant states in space are, in the forward direction (left to right) the "dimerised state" obtained from the tensor product of (with appropriate boundaries, see below), and in the backward direction (right to left) the flat state. This is consistent with the space dynamics underŨ being (right) stochastic, and is reminiscent of dual-unitarity [50,76,77] (see also e.g., [49,57,[78][79][80][81][82][83][84][85][86][87][88][89]) in the quantum setting. ...

We study the dynamics of a classical circuit corresponding to a discrete-time deterministic kinetically constrained East model. We show that -- despite being deterministic -- this "Floquet-East" model displays pre-transition behaviour, which is a dynamical equivalent of the hydrophobic effect in water. By means of exact calculations we prove: (i) a change in scaling with size in the probability of inactive space-time regions (akin to the "energy-entropy" crossover of the solvation free energy in water), (ii) a first-order phase transition in the dynamical large deviations, (iii) the existence of the optimal geometry for local phase separation to accommodate space-time solutes, and (iv) a dynamical analog of "hydrophobic collapse". We discuss implications of these exact results for circuit dynamics more generally.

... Simple model systems that can be solved analytically are highly desirable since they offer an invaluable window into these questions. In recent years dualunitary circuits (DUCs) have emerged as paradigmatic examples of exactly solvable yet chaotic many-body systems [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] in which a variety of dynamical quantities are analytically accessible [10][11][12][13][14][15]17,18]. However, solvability comes at a cost in genericity, and in several respects DUCs display behavior that differs strikingly from the phenomenology observed numerically in more generic models [4][5][6][7][8]28,29]. ...

Dual-unitary circuits are paradigmatic examples of exactly solvable yet chaotic quantum many-body systems, but solvability naturally goes along with a degree of nongeneric behavior. By investigating the effect of weakly broken dual unitarity on the spreading of local operators, we study whether, and how, small deviations from dual unitarity recover fully generic many-body dynamics. We present a discrete path-integral formula for the out-of-time-order correlator and recover a butterfly velocity smaller than the light-cone velocity, vB<vLC, and a diffusively broadening operator front, two generic features of ergodic quantum spin chains absent in dual-unitary circuit dynamics. The butterfly velocity and diffusion constant are determined by a small set of microscopic quantities, and the operator entanglement of the gates has a crucial role.

... These circuits are characterized by an underlying spacetime duality, resulting in unitary dynamics in both time and space. This duality allows for exact calculations of the dynamics of correlation functions of local observables [9,[11][12][13][14], out-of-time-order correlators [15,16], entanglement [10,17], indicators of quantum chaos [18][19][20][21], and dual-unitary circuits have since been the subject of active theoretical [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38] and experimental studies [39,40]. ...

Dual-unitary circuits have emerged as a minimal model for chaotic quantum many-body dynamics in which the dynamics of correlations and entanglement remains tractable. Simultaneously, there has been intense interest in the effect of measurements on the dynamics of quantum information in many-body systems. In this work we introduce a class of models combining dual-unitary circuits with particular projective measurements that allow the exact computation of dynamical correlations of local observables, entanglement growth, and steady-state entanglement. We identify a symmetry preventing a measurement-induced phase transition and present exact results for the intermediate critical purification phase.

... For example, these have been used for the evaluation of dynamical correlation functions [3,5], spectral statistics [4], construction of a quantum ergodic hierarchy [11], entanglement generation [7], exact emergence of random matrix universality [13], and measurement induced phase-transitions [14]. It has been shown to be classically simulatable for short times or circuit depths for certain initial states and for local expectation values [9]. However, for late times the problem has been shown to be BQP-complete and the dual-unitary circuits are capable of universal quantum computation, while classical simulation of the problem of sampling has been shown to be hard [9]. ...

... It has been shown to be classically simulatable for short times or circuit depths for certain initial states and for local expectation values [9]. However, for late times the problem has been shown to be BQP-complete and the dual-unitary circuits are capable of universal quantum computation, while classical simulation of the problem of sampling has been shown to be hard [9]. ...

... Unlike the previous example, S R α is full-rank (except for α = 0, with the largest singular value being distinct and the others are equal. As the largest singular value corresponds to the maximally entangled standard Bell state, we get that the lower bound is still exact and K D (S α ) = 2d 2 − 2d d 2 cos 2 (πα/2) + sin 2 (πα/2), (9) which interpolates between 0, when α = 0, corresponding to the identity operator, to the maximum K * D when α = 1 and the swap gate is dual unitary. Although in the examples above K D (U) = K * D (U), it is important to note that such an equality does not hold in general. ...

The problem of finding the resource free, closest local unitary, to any bipartite unitary gate is addressed. Previously discussed as a measure of nonlocality, and denoted $K_D(U)$ , it has implications for circuit complexity and related quantities. Dual unitaries, currently of great interest in models of complex quantum many-body systems, are shown to have a preferred role as these are maximally and equally away from the set of local unitaries. This is proved here for the case of qubits and we present strong numerical and analytical evidence that it is true in general. An analytical evaluation of $K_D(U)$ is presented for general two-qubit gates. For arbitrary local dimensions, that $K_D(U)$ is largest for dual unitaries, is substantiated by its analytical evaluations for an important family of dual-unitary and for certain non-dual gates. A closely allied result concerns, for any bipartite unitary, the existence of a pair of maximally entangled states that it connects. We give efficient numerical algorithms to find such states and to find $K_D(U)$ in general.