Sequential generation of projected entangled-pair states

Tensor network states play a fundamental role both in quantum information processing and many-body physics, as they are natural representations of states with area-law entanglement Verstraete et al. (2008); Orús (2019); Cirac et al. (2020). In one dimension, Matrix-Product States (MPS) Fannes et al. (1992); Perez-Garcia et al. ; Schollwöck (2011) efficiently approximate the ground state of gapped Hastings (2007) and critical Hamiltonians Verstraete and Cirac (2006). Their higher-dimensional generalizations, Projected Entangled-Pair States (PEPS) Verstraete and Cirac , also play an important role in many-body physics. Apart from providing efficient approximations in different scenarios, they embrace many paradigmatic states of condensed matter physics, including topological states like the toric code Verstraete et al. (2006); Schuch et al. (2010) and string-net states Levin and Wen (2005); Gu et al. (2009); Buerschaper et al. (2009), or resonating valence bound states Verstraete et al. (2006). They also contain elements that are relevant in the context of quantum metrology Degen et al. (2017), like the W Dür et al. (2000) or GHZ states Greenberger et al. (1989), or in quantum computing, like the cluster Briegel and Raussendorf (2001), graph Hein et al. (2004); Russo et al. (2019); Azuma et al. (2015) and hypergraph state Kruszynska and Kraus (2009); Rossi et al. (2013). Thus, the efficient preparation of such states would have an important impact on the study of many-body systems and quantum information.

One can generate MPS by sequentially applying local unitaries Schön et al. (2005, 2007), which provides a way to deterministically prepare entangled states on quantum computers Smith et al. ; Lin et al. (2021) or in photonic systems Gheri et al. (1998); Saavedra et al. (2000); Schön et al. (2007); Lindner and Rudolph (2009); Schwartz et al. (2016); Tiurev et al. ; Eichler et al. (2015); Schwartz et al. (2016); Besse et al. (2020); Wei et al. (2021a); Wein et al. , with a generation time (circuit depth) that scales linearly with the system size $n$ (number of qudits) as $O(n)$. Moreover, sequential MPS generation is an essential component in numerous theoretical frameworks Lamata et al. (2008); Cramer et al. (2010); Saberi (2011); Delgado et al. (2007); Osborne et al. (2010); Huggins et al. (2019); Foss-Feig et al. ; Barratt et al. (2021); Ran (2020); Lin et al. (2021).

Efficient generation of PEPS is, however, much more difficult. Even in two dimensions, it is believed that most states will require a preparation time that increases exponentially with the system size Verstraete et al. (2006); Schuch et al. (2007). Nevertheless, most of the paradigmatic examples mentioned above can be efficiently prepared also in higher dimensions, and experimental efforts have already started Satzinger et al. ; Liu et al. (2021). This calls for efforts to identify, classify, and extend subclasses of PEPS that allow for efficient preparation, ideally together with an explicit algorithm to do so. In this vein, there are two subclasses of PEPS in two dimensions that stand out: (i) sequentially generated states (SGS) Bañuls et al. (2008), and (ii) PEPS generated by photon feedback (F-PEPS) Pichler et al. (2017). Interestingly, both of these classes can be obtained from a product state by a sequential quantum circuit.

In this paper, we introduce plaquette PEPS (P-PEPS), which are defined by sequentially applying unitaries to plaquettes of qudits initially in a product state. P-PEPS can straightforwardly be expressed as PEPS and naturally encompass SGS and F-PEPS. We focus on a particular radial plaquette ordering, which leads to a subclass we call radial plaquette PEPS (RP-PEPS). This class allows certain local observables to be computed efficiently and has SGS and isometric tensor network states (isoTNS) Zaletel and Pollmann (2020) as proper subclasses. Our construction thus provides a quantum circuit to prepare isoTNS, which is a class that has been shown to include graph states and hypergraph states of local connectivities, and all string-net states Soejima et al. (2020). While for a $q$-dimensional lattice of $N=n_{1}\times...\times n_{q}$ sites, in the worst case, P-PEPS require a circuit depth scaling with the total number of sites, RP-PEPS can be prepared particularly efficiently, with the circuit depth $T_{\rm rp}$ scales as the side length of the lattice

We also show that an array of coupled quantum sources each comprising an ancilla–emitter pair can naturally produce RP-PEPS of flying qub(d)its with the same efficient scaling, and prepare F-PEPS with a circuit depth $O(N)$. This includes a wide variety of high-dimensional states that have been proposed for sequential photon generation Economou et al. (2010); Gimeno-Segovia et al. (2019); Russo et al. (2019); Bekenstein et al. (2020); Pichler et al. (2017); Dhand et al. (2018); Xu and Fan (2018); Wan et al. ; Zhan and Sun (2020); Shi and Waks ; Bartolucci et al. ; Bombin et al. ; Wei et al. (2021a). Overall, P-PEPS (RP-PEPS) and their generation protocols apply to photonic systems, and to platforms with matter qub(d)its like superconducting circuits Blais et al. (2021), trapped ions Bruzewicz et al. (2019), or Rydberg atoms Saffman (2016), where local interactions can be engineered with high precision.

Plaquette PEPS.—For concreteness, we restrict our attention to a two-dimensional lattice of qudits of size $N=n\times m$, and the high-dimensional generalization will be discussed in SI ¹¹1See Supplemental Material for more details, which includes Refs.[74,75].

We define P-PEPS with periodic boundary conditions (PBC) as the states generated from the product state $|0\rangle^{\otimes N}$ through sequential application of unitaries to plaquettes of size ${L_{p}}\times{L_{p}}$ (${L_{p}}\ll m,n$) [c.f. Fig. 1(a)]

where $\vec{v}_{\mu}=(i_{\mu},j_{\mu})$ and the unitary $\hat{U}_{\vec{v}_{\mu}}$ acts on qudits in the square spanning from $(i_{\mu},j_{\mu})$ to $(i_{\mu}+L_{p}-1,j_{\mu}+L_{p}-1)$, and we identify the rows $i\pm m\equiv i$ and columns $j\pm n\equiv j$. Here the choice of plaquette shape reflects the locality of the unitaries. The ordering of the unitaries ${\cal P}=\left({{{\vec{v}}_{1}},{{\vec{v}}_{2}},...,{{\vec{v}}_{N}}}\right)$ fulfils the conditions $\vec{v}_{\mu}\neq\vec{v}_{\nu}$ for $\mu\neq\nu$. We show an example of $\cal P$ in Fig. 1(b), and call the position of the first unitary $\vec{v}_{1}$ the source point. To define the state with open boundary conditions (OBC), we simply omit gates that act across boundaries.

P-PEPS is a subclass of PEPS, which are states defined through a network of tensors with one tensor per lattice site [Fig. 1(c)], whose virtual indices are contracted with their neighbors. In two dimensions,

where ${B_{[{i,j}]}^{k}}_{lurb}$ is a rank-5 tensor on the site $(i,j)$ that has one physical index $k$ of dimension $d$ and four virtual indices $l,u,r,b$ of bond dimension $D$. The symbol $\cal F_{\rm 2D}$ denotes the contraction of all virtual indices. To obtain the PEPS representation of P-PEPS, we decompose the plaquette unitaries into projected entangled-pair operators (PEPO) Cirac et al. (2020) [c.f. Fig. 1(d)]. This allows one to write the whole sequential circuit as a PEPO, which, applied to a product state, yields a PEPS with bond dimension $D\leq O(d^{L_{p}^{4}})$ Note (1).

We are particularly interested in cases where each unitary overlaps with at least one of the earlier ones, such that they create correlations. The state shown in Fig. 1(a) is such an example. Sequential circuits with overlapping unitaries efficiently establish correlations between arbitrary locations of the lattice with $O(N)$ unitaries Note (1). This should be contrasted with brickwall circuits Gopalakrishnan and Lamacraft (2019) that take $O(N\cdot\mathop{\max}\limits_{i}{n_{i}})$ unitaries to do so Note (1). This implies that P-PEPS offers a more efficient parametrization of states with correlations across the entire system. Moreover, while P-PEPS have area-law entanglement, brickwall circuits that create long-range correlations will instead lead to states with volume-law entanglement Nahum et al. (2017, 2018); Von Keyserlingk et al. (2018).

Radial plaquette PEPS (RP-PEPS).— Naively, it takes a circuit depth $O(N)$ to create a P-PEPS [c.f.  Eq. 2]. However, some orderings $\cal P$ allow unitaries to be applied in parallel. A simple example is to arrange the unitaries as a brickwall circuit of depth $O(L_{p}^{2})$. Here we define a subclass of P-PEPS, RP-PEPS, where starting from the source point, the positions $\{\vec{v}_{\mu}\}$ of the unitaries are ordered such that they can be grouped to multiple layers of commuting unitaries, and each layer act on the boundary of the existing gate-acted region. An example with $L_{p}=2$ is illustrated in Fig. 1(e), where the gates are grouped as $\left[{\left(1\right),\left({2,3}\right),\left({4\textrm{--}7}\right),\left({8\textrm{--}13}\right),\left({14\textrm{--}21}\right),...}\right]$ (denoted by shades of different colors). To resolve ambiguities in the plaquette order, we choose preferred directions in which the position of the plaquette moves. In Fig. 1(e) we choose ‘horizontal first, and positive direction first’. The circuit depth of preparing RP-PEPS is asymptotically ${T_{{\rm{rp}}}}\approx n+{L_{p}}m$, following the scaling in Eq. 1. Moreover, RP-PEPS allow efficient computation of expectation values of local observables that are geometrically close to the source point or the line that passes through the source point along the preferred direction, and this generically implies that correlation functions in these regions decay exponentially. This is reminiscent of isoTNS Zaletel and Pollmann (2020).

The above definitions straightforwardly generalize to higher dimensional lattices, where plaquettes become high-dimensional cubes Note (1). While the general circuit depth for P-PEPS again scales with $N$, RP-PEPS obeys Eq. 1.

Relation to other families of PEPS.—By definition, P-PEPS can be efficiently prepared, have a PEPS description, and host long-range correlations. Now we show that P-PEPS naturally encompass other families of PEPS that are prepared sequentially (SGS and F-PEPS), as well as isoTNS (we follow the definition in Ref. Zaletel and Pollmann (2020), and see Ref. Haghshenas et al. (2019) for a different definition).

SGS [c.f. Fig. 2(a)] are defined in terms of linear sequential circuits comprising unitaries $\{\hat{V}_{[i,j]}\}$ of length $L_{p}$ acting on rows across qudits whose columns have been prepared in MPS Bañuls et al. (2008)

The MPS in each column can be put in canonical form, such that they can be written as linear sequential circuits Schön et al. (2005); Bañuls et al. (2008). This allows us to identify the tensor of the corresponding PEPS as two overlapping $L_{p}$-qudit unitaries [c.f. Fig. 2(b)]. These two unitaries are contained in a $L_{p}\times L_{p}$ plaquette unitary. Thus, each SGS can be written as an RP-PEPS, with the source point at the bottom left of the lattice in the case of Fig. 2(a).

To be precise, let us denote the class of SGS (P-PEPS) on an $n\times m$ lattice with circuit length (plaquette length) $L_{p}$ as ${\rm SGS}_{n\times m}^{L_{p}}$ (${\textrm{P-PEPS}}_{n\times m}^{L_{p}}$), we have

The tensors of SGS in the bulk satisfy an isometry condition shown in Fig. 2(d), which is the same condition as is obeyed by the tensors in isoTNS Zaletel and Pollmann (2020). Indeed, as we show in the following, these classes are closely related.

IsoTNS are PEPS [Eq. 3] in which all tensors satisfy isometry conditions that depend on their position in the lattice. Specifically, when all incoming indices of a tensor (denoted by incoming arrows in Fig. 2c) and the physical index are contracted with corresponding indices of the complex conjugate of that tensor, the remaining indices yield the identity. For example, the tensor in the dashed box in Fig. 2(c) obeys [c.f. Fig. 2(d)]

The red shaded lines in Fig. 2(c) are called orthogonality hypersurfaces, which only have incoming arrows, and their intersection is the orthogonality center (OC) Zaletel and Pollmann (2020).

One can prepare isoTNS as RP-PEPS, but restricting the unitaries in the bulk to be ‘L’-shaped, as shown in Fig. 3(a1-6). The required three-qudit unitaries can be written as

where $|k,r,u\rangle\equiv|{k_{\left[{i,j}\right]}},{r_{\left[{i,j+1}\right]}},{u_{\left[{i+1,j+1}\right]}}\rangle$, and the tensor ${{B_{[{i,j}]}^{k}}_{lurb}}$ automatically satisfies the isometry condition Eq. 6. Thus each unitary creates an isoTNS site [c.f. Fig. 3(b)].

Sequentially applying the gates shown in Fig. 3(a1-6) gives rise to the tensor contraction pattern shown in Fig. 3(a7), which represents an arbitrary isoTNS with OC in the corner. The generated isoTNS has bond dimension $d$ and physical dimension $d$, except at the right boundary, where two sites of each row are combined to form a site with physical dimension $d^{2}$. Note that arbitrary isoTNS of that geometry with a uniform physical dimension can be embedded in that state by setting the rightmost qudits to zero and treating them as ancillas. Moreover, it is clear from Fig. 3(a) that the circuit depth for preparing isoTNS is ${T_{{\rm{iso}}}}\approx n+m$. We show in the SI Note (1) that: i) by extending the indices of ‘L’-shaped unitaries to $2s+1$ qudits with $s=\lceil{{{\log}_{d}}D}\rceil$ and changing the source point of the RP-PEPS, isoTNS with arbitrary bond dimension $D$ and with OC in the bulk can be prepared. ii) This protocol can be generalized to prepare isoTNS of higher dimensions. Therefore, isoTNS on arbitrary lattices (of size ${n_{1}}\times...\times{n_{q}}$) admit exact representations as sequential quantum circuits, with the circuit depth scaling as

The above observation shows that isoTNS $\subset$ RP-PEPS. A similar relation also holds between P-PEPS and F-PEPS Pichler et al. (2017), here understood as a generalization to qudits and with arbitrary photon feedback. F-PEPS can be viewed as isoTNS on a lattice with different connectivity Pichler et al. (2017). If we denote ${\rm{isoTNS}}_{n\times m}^{D,d}$ (${\textrm{F-PEPS}}_{n\times m}^{D,d}$) as the class of isoTNS (F-PEPS) on a $n\times m$ lattice with bond dimension $D$ and physical dimension $d$, we prove in the SI Note (1) that isoTNS (F-PEPS) is contained in RP-PEPS (P-PEPS) with a slightly larger lattice

Having established that both SGS and isoTNS are RP-PEPS with ‘L’-shaped unitaries, we note that SGS has a further condition on the unitaries, namely that they can be decomposed into two unitaries corresponding to the two arms of the L (see Fig. 2b), which indicates that SGS are a subclass of isoTNS. This has direct consequences for the states. While in SGS, local observables can efficiently be calculated anywhere in the lattice, in isoTNS, this requires shifting the OC, which can only be done approximately. Their precise relation is Note (1)

Finally, we note that, since the isometry of the PEPS tensors derives directly from the unitarity of the preparation circuit, we further show that RP-PEPS can be expressed as isoTNS on lattices with unusual connectivities Note (1).

Generating RP-PEPS of flying qub(d)its.— Ref. Schön et al. (2005) introduces a protocol to prepare arbitrary photonic MPS of bond dimension $D$ and physical dimension $d$ using a photon source comprising a $D$-level ancilla $A_{1}$ and a $d$-level emitter $E_{1}$. The MPS is prepared by repeatedly applying a unitary on the joint ancilla-emitter system, followed by swapping the emitter state into a flying photon, defined in terms of the photon emission isometry $M_{\rm ph}$

Now we extend the above protocol by considering an array of $m$ sequential photon sources coupled to each other as shown in Fig. 4, and show that photonic RP-PEPS can be prepared with this setup.

The protocol is shown in Fig. 4. Here we assume each ancilla has a dimension $D=d^{L_{p}-1}$, so it can be thought of as $L_{p}-1$ qudits. Starting with the ancillas $\{A_{j}\}$ and emitters $\{E_{j}\}$ in their ground state $|{{\varphi_{0}}}\rangle=|{{0_{\{{{A_{j}}}\}}}}\rangle\otimes|{{0_{\{{{E_{j}}}\}}}}\rangle$, in the step to prepare the $(i,j)$-th site of the RP-PEPS, first we apply a unitary ${\hat{U}_{[{i,j}]}}$ that acts on the ancillas $\{A_{j},...,A_{j+L_{p}-1}\}$ and emitters $\{E_{j},...,E_{j+L_{p}-1}\}$ [see Fig. 4(a) for $L_{p}=2$ case]. After the unitary, we trigger the photon emission from the emitter $E_{j}$ (denoted as $M^{j}_{{\rm{ph}}}$ [cf. Eq. 12]). To disentangle the ancilla from the photons, in the last $L_{p}-1$ steps of the protocol, we sequentially swap the effective $L_{p}-1$ qudits contained in the ancilla into photons (c.f. Fig. 4b), an operation collectively denoted by $\cal S$. The final state of the system is $\left|{{\varphi_{0}}}\right\rangle{\left|{{\psi_{\rm rp}}}\right\rangle_{\rm ph}}$, with the photonic state

$|\psi_{{\rm{rp}}}{\rangle_{{\rm{ph}}}}$ is an arbitrary RP-PEPS with OBC and plaquette size $L_{p}$, with its source point at the first photonic qudit. The circuit depth is the same as that of matter-based lattice case [c.f. Eq. 1]. The same protocol also allows to prepare isoTNS Note (1), and this setup can be used to prepare photonic F-PEPS with circuit depth $O(N)$ Note (1).

Notice that, at the boundary of the photon source array, the photon emission process emits multiple photons, as visualized in Fig. 4(a4). In contrast to the protocol that produces RP-PEPS on a matter-based lattice, here the overlapping of gates along the horizontal direction are produced by acting on the ancillas. In Ref. Wei et al. (2021b) we propose a cavity-transmon setup to realize of this protocol, and elaborate on how to create the two-dimensional photonic cluster state and the toric code state ²²2Note that for the cluster state generation, there are specific schemes Lindner and Rudolph (2009); Economou et al. (2010) that do not need ancillas and SWAP gates. These schemes thus require less resources than the present scheme..

Conclusion.—We have introduced P-PEPS and its subclass RP-PEPS, which constitute a natural generalization of sequential preparation protocols from one to higher dimensions. These states satisfy area-law entanglement by construction, combine the capacity to host long-range correlations, topologically ordered states, and a large subclass of PEPS with a simple and efficient preparation protocol. Our work helps to clarify the relation between various relevant classes of PEPS, including SGS Bañuls et al. (2008), F-PEPS  Pichler et al. (2017) and isoTNS Zaletel and Pollmann (2020), that we show SGS $\subset$ isoTNS $\subset$ RP-PEPS, and F-PEPS $\subset$ P-PEPS.

The family of states we introduce come with explicit protocols that prepare them in matter-based and photon-based lattices, which makes them promising targets for near-term experimental realization. Furthermore, one can include several layers of sequential plaquettes to increase the expressivity of the ansatz.

We acknowledge funding from ERC Advanced Grant QUENOCOBA under the EU Horizon 2020 program (Grant Agreement No. 742102), and within the D-A-CH Lead-Agency Agreement through project No. 414325145 (BEYOND C), and the European Union’s Horizon 2020 research and innovation program under Grant No. 899354 (FET Open SuperQuLAN).

Supplementary Materials