Building spatial symmetries into parameterized quantum circuits for faster training

In this work, we consider a generic learning task ubiquitous to the field of VQAs: given a Hamiltonian H, one is interested in preparing its ground state. In the context of VQAs, this task is formulated as an optimization over states $\vert \psi(\boldsymbol{\theta})\rangle$ that are realized by applying a parameterized unitary $U(\boldsymbol{\theta})$—implemented by means of a quantum circuit—to a fixed initial state $\vert \psi_0\rangle$. The circuit parameters $\boldsymbol{\theta}$ are then optimized to minimize the energy of the realized state, that is, the expectation value

$$\begin{equation} E\left(\boldsymbol{\theta}\right) = \langle\psi\left(\boldsymbol{\theta}\right)|H|\psi\left(\boldsymbol{\theta}\right)\rangle = \langle\psi_{0}|U\left(\boldsymbol{\theta}\right)^{\dagger} H U\left(\boldsymbol{\theta}\right)|\psi_{0}\rangle \,. \end{equation} \tag{ 1 }$$

2.1.1. Circuit structure

At the core of the success of any task of ground state preparation lies the structure of the circuit employed. Quantum circuits are typically structured as a composition of repeated layers

$$\begin{equation} U\left(\boldsymbol{\theta}\right) = \prod^L_{l = 1} V\left(\boldsymbol{\theta}^{\left(l\right)}\right)\, \end{equation} \tag{ 2 }$$

where V denotes a parametrized unitary corresponding to the action of a layer, with parameters $\boldsymbol{\theta}^{(l)}$ varying from one layer (indexed as $l \in [1,L]$) to another. These layers can be further decomposed in terms of gates, typically acting on single qubits or two qubits, that are either fixed gates or parameterized ones of the form $R_G(\theta) = \exp[-i \theta G]$ for a generator G (a traceless Hermitan operator) and a variable parameter θ. We denote as $\mathcal{G} = \{G_g\}$ the set of all the generators involved in a circuit.

2.1.2. HEA circuits

In the most general case, the parameters associated to each of the parameterized gates involved in a given layer can be taken to be uncorrelated. For instance, the HEA ansatz employed in [5] has layers of the form

$$\begin{equation} V_{\text{HEA}}\left(\boldsymbol{\theta}^{\left(l\right)}\right) = U_{ent} \times \prod^n_{i = 1} R_{Z}\left(\boldsymbol{\theta}_{i,1}^{l}\right) R_{X}\left(\boldsymbol{\theta}_{i,2}^{l}\right) R_{Z}\left(\boldsymbol{\theta}_{i,3}^{l}\right) \end{equation} \tag{ 3 }$$

composed of a fixed entangling unitary U_ent and parametrized single-qubit gates taken to be either Pauli X_i or Pauli Z_i rotations acting on the qubits $i = 1, \ldots, n$. Notably, each of these 3n rotations per layer is controlled by an individual parameter $\boldsymbol{\theta}_{i,o\in \{1,2,3\}}^{(l)}$ that can be tuned independently of any other rotation. In such a case, neither the choice of the gates nor the choice of the parameter structure depends on the underlying problem to be solved.

2.1.3. HVA-like circuits

As an alternative paradigm, in the circuit layers of the HVA-like anstazes (encompassing HVA circuits [10–12], QAOA circuits [2] and their generalization [14]) most of the parameters are correlated. That is, a single parameter value is used for many of the gates appearing in the same layer. For instance, a layer of the the alternating ansatz in [10–12] takes the form of

$$\begin{equation} V_{\text{HVA}}\left(\boldsymbol{\theta}^{\left(l\right)}\right) = \prod^n_{i = 1} R_{X_i}\left(\boldsymbol{\theta}_{1}^{\left(l\right)}\right) \prod_{\left(i,j\right) \in \mathcal{I}} R_{Z_i Z_j}\left(\boldsymbol{\theta}_{2}^{\left(l\right)}\right), \end{equation} \tag{ 4 }$$

where each of the n qubits are rotated along the X–axis with the same angle $\boldsymbol{\theta}_{1}^{(l)}$, and where each pairs of qubits (i, j) belonging to some set $\mathcal{I}$, defined in correspondence to an underlying Hamiltonian, are subjected to ZZ interactions with the same strength $\boldsymbol{\theta}_{2}^{(l)}$. Going beyond these two extreme cases of independent and (almost fully) correlated parameters is the topic of this work.

Following the many proposals of VQAs ansatzes, it has been realized that their application to large problem sizes would, in some cases, quickly result in trainability issues rendering the optimization of the circuit parameters most likely to fail. In particular, the phenomenon of barren plateaus (BPs) [6], or exponentially vanishing gradients, even in its most benign form, greatly limits the extent to which quantum circuits can be trained at scale. BPs result from too expressive anstazes [6–9], the usage of global cost functions [7], the generation of entanglement [27, 28], or the presence of noise [15, 29].

Recall that BPs are defined as an exponential decay of the magnitude of the circuit gradients with n. We denote by

$$\begin{equation} \partial_i E \equiv \partial E\left(\boldsymbol{\theta}\right) / \partial \boldsymbol{\theta}_i \end{equation} \tag{ 5 }$$

the ith element of the gradient vector of the cost $E(\boldsymbol{\theta})$ in equation (1), and by $\text{Var}( \partial_i E)$ the variance of such elements obtained over random circuit parameters. BPs occur when $\text{Var}( \partial_i E)\in \mathcal{O}(1/\mathrm{exp}(n))$, that is, when the partial derivatives concentrate exponentially in the system size. In practice, values of the cost and its gradients are estimated with an error ${\sim}1/\sqrt{N}$ for a number of measurement repetitions N. Hence, in the case of BPs, N would need to scale exponentially in order to distinguish true values of the gradients or cost function differences from statistical noise [30]. This would render optimization unfeasible beyond some critical system size. In the following we review how such BPs, particularly those caused by expressibility and those caused by noise, connect to the design of quantum circuits.

2.2.1. Expressibility and the Dynamical Lie Algebra

Circuits acting on n qubits realize unitaries belonging to $\mathbb{U}(d)$, the group of unitaries of dimension d. For a given choice of circuit layout, it is of interest to understand what subset $U(\boldsymbol{\theta}) \subseteq \mathbb{U}(d)$ can be reached when varying the values of the parameters $\boldsymbol{\theta}$, i.e. to understand how expressive a circuit is. To this intent, several metrics have been proposed [9, 31, 32]. Here, we restrict our attention to the dimension of the circuit's Dynamical Lie Algebra (DLA), a measure originating from the field of algebraic quantum optimal control [33] that was revisited in the context of VQAs [9, 32]. We now briefly review such concept, and refer the reader to [9] for a more detailed exposition.

Starting from the generator set $\mathcal{G}$ of a given circuit, one can define its DLA (denoted $\mathfrak{g}$), as the vector space spanned by the individual generators and their repeated nested commutators:

$$\begin{equation} \mathfrak{g} = \text{Span} \langle i \mathcal{G} \rangle_{\mathrm{Lie}}, \end{equation} \tag{ 6 }$$

where $\langle \cdot \rangle_\textrm{Lie}$ denotes the Lie closure, the set generated by repeated nested commutators of the set. In turn, it can be shown [33] that unitaries realized by parametrized quantum circuits with a generator set $\mathcal{G}$, are of the form $U(\boldsymbol{\theta}) = e^{g}$ for some $g \in \mathfrak{g}$. That is, the Lie algebra $\mathfrak{g}$ fully characterizes the (continuous Lie group of) unitaries that can be realized by the circuit.

Let us stress that the ability of $U(\boldsymbol{\theta})$ to realize exactly $U = e^{g}$, for any $g \in \mathfrak{g}$, assumes a sufficiently large number of layers. Still, understanding circuits' expressibility through their DLA bears a particular appeal for our purposes. First, great progress has been made in characterizing DLAs of some common circuit constructions [34]. Second, it has recently been conjectured, and verified in several examples, that $\text{Var}( \partial_i E) \in \mathcal{O}(1/ \mathrm{\operatorname{poly}}(\text{dim}(\mathfrak{g})))$ [9, 35]. In other words, BPs can arise as a consequence of a DLA that scales exponentially with n, as is the case of HEA circuits. Given such connection to circuit trainability, we will resort to DLA characterization when assessing expressibility of circuits later on.

2.2.2. Noise

A symmetry S of a Hamiltonian H is a unitary operator ($SS^{\dagger} = S^{\dagger}S = I$) leaving H invariant, such that $S H S^{-1} = H$ (or equivalently, such that $[S, H] = 0$). In cases where H admits more than one symmetry (the identity operation being a trivial one), these organize as groups. In particular, given two symmetries S₁ and S₂ their compositions ($S_1S_2$ or $S_2S_1$) and their inverses ($S_1^{-1}$ and $S_2^{-1}$) are also symmetries.

Symmetries of quantum systems can be broadly classified in two classes. On one hand, spatial symmetries (or outer symmetries [33]) are those symmetries which are permutations of sites of the system. These form a discrete group of symmetries, with order as large as $n!$ for the case of the Symmetric group S_n , and will be the symmetries of particular interest for this work. Not all symmetries fall in this first category and the remaining ones are termed internal symmetries (or inner ones). These can either be discrete groups, as is the case of $\mathbb{Z}_2 = \{I, X^{\otimes n}\}$, or continuous ones, as is the case of $\mathbb{SU}_2 = \{U^{\otimes n}|U \in \text{SU(2)}\}$. Designing quantum circuits that respect such inner symmetries is usually achieved by a choice of gate generators commuting with them, e.g. $\mathcal{G}_1 = \{\text{SWAP}_{i,j}\}$ with $\text{SWAP}_{i,j}$ the swap operator between qubit i and j for the case of $\mathbb{SU}_2$.

Turning back our focus to spatial symmetries, and considering a n–qubit system, we denote by $\pi \in S_n$ the permutations over the set of n indices that label the qubits. The action of π on a n–qubit state $\vert \psi\rangle \rightarrow \pi \vert \psi\rangle$ is understood as the corresponding permutation of its qubit indices (note the overloading of the symbol π to a unitary operator when acting on states). For instance, for a state $\vert \psi\rangle = \vert s_1\rangle\vert s_2\rangle\vert s_3\rangle$ and $\pi(1) = 2$, $\pi(2) = 1$ and $\pi(3) = 3$ (i.e. a permutation of the indices 1 and 2), then $\pi \vert \psi\rangle = \vert s_2\rangle\vert s_1\rangle\vert s_3\rangle$. Accordingly, π acts on operators as $O \rightarrow \pi O \pi^{-1}$.

We call the spatial symmetries of a Hamiltonian its automorphism group, that is defined as follows.

Definition 1. Given a Hamiltonian H, we define its automorphism group $\mathrm{Aut}(H) \subset S_n$ as the subgroup of permutations π_a that leave H unchanged.

$$\begin{equation} \mathrm{Aut}\left(H\right) = \left\{\pi_a \in S_n | \pi_a H \pi_a^{-1} = H\right\}. \end{equation} \tag{ 7 }$$

Given commutation of H with an element $\pi_a \in {\mathrm{Aut(H)}}$, it is known that H can be simultaneously diagonalized with π_a , that is, H is block diagonal in the eigenbasis of π_a . More generally, operators H commuting with groups of unitaries, as in equation (7), will also adopt a block decomposition inherited from the structure of the group. For instance, in the case of translational invariance of a system, each of these blocks corresponds to a subspace of states with given momentum. Crucially, the search for a ground state can be limited to a search within each (or a subset) of the individual symmetry blocks, thus reducing the Hilbert space that has to be considered and the expressibility of the circuits employed. Given that circuits with the same symmetries also share the same block structure, when applied to an initial state with support in a single block, one can ensure that these will prepare states remaining in the same block, thus allowing exploration of a smaller subspace of the Hilbert space.

In the following, we focus our attention on ground-state preparations in the invariant subspace of ${\mathrm{Aut}(H)}$ which contains all the states $\vert \psi(\boldsymbol{\theta})\rangle$ such that for all $\pi_a \in {\mathrm{Aut}(H)}$, $\vert \psi(\boldsymbol{\theta})\rangle = \pi_a \vert \psi(\boldsymbol{\theta})\rangle$. As we now discuss, search in this subspace can be efficiently parameterized by equivariant circuits applied to easy-to-prepare initial states such as $\vert 0\rangle^{\otimes n}$ or $\vert +\rangle^{\otimes n}$. Note however that the same techniques are directly applicable to any other block defined by ${\mathrm{Aut}(H)}$, as long as the corresponding initial states can be efficiently prepared.

Here, we show that given (i) an initial state invariant under permutations, and (ii) a circuit equivariant with respect to ${\mathrm{Aut}(H)}$ one can ensure that the states prepared by the quantum circuit are invariant under ${\mathrm{Aut}(H)}$.

Definition 2. A circuit $U(\boldsymbol{\theta})$ is said to be equivariant with respect to a symmetry group $\mathbb{G} \subset S_n$ whenever

$$\begin{equation} \forall \pi \in \mathbb{G},\; U\left(\boldsymbol{\theta}\right) = \pi U\left(\boldsymbol{\theta}\right) \pi^{-1}\ . \end{equation} \tag{ 8 }$$

We denote the space of such circuits as $\mathcal{E}(\mathbb{G})$.

Given $U(\boldsymbol{\theta}) \in \mathcal{E}({\mathrm{Aut}(H)})$ and a symmetric input state $\vert \psi_0\rangle$ (e.g. $\vert 0\rangle^{\otimes n}$ or $\vert +\rangle^{\otimes n}$), it can be verified that, as desired, the realized state is invariant under any $\pi_a \in {\mathrm{Aut}(H)}$:

$$\begin{align} \begin{split} \vert \psi\left(\boldsymbol{\theta}\right)\rangle & = U\left(\boldsymbol{\theta}\right) \vert \psi_0\rangle = \pi_a U\left(\boldsymbol{\theta}\right) \pi_a^{-1}\vert \psi_0\rangle \\ & \pi_a U\left(\boldsymbol{\theta}\right)\vert \psi_0\rangle = \pi_a \vert \psi\left(\boldsymbol{\theta}\right)\rangle. \end{split} \end{align} \tag{ 9 }$$

Note that the requirement of a symmetric initial state could be relaxed to a requirement of an initial state invariant under the automorphism group only. However, this may result in input states that are hard to prepare in practice.

The equivariance property of unitaries as defined in definition 2 is associative by composition. That is, given U₁ and $U_2 \in \mathcal{E}({\mathrm{Aut}(H)})$, their product, $U_1U_2$ or $U_2U_1$, also belongs to $\mathcal{E}({\mathrm{Aut}(H)})$). As such, imposing equivariance of a circuit can be achieved in a constructive manner by imposing equivariance at its constituent level, i.e. at the level of its layers.

As we show in the next section, HVA-like circuits naturally enforce such equivariance properties, thus effectively limiting their expressibility and improving their trainability. This, however, is achieved at the cost of reducing greatly the number of independent parameters per layer. Still, they can be readily generalized to ORB circuits that preserve equivariance, but also, that maximize the number of free parameters per layer.

Let us begin by discussing HVA-like circuits. As mentioned earlier, and exemplified in equation (4), HVA-like circuits are most often composed of 1- and 2-qubit gates with shared parameters.

3.3.1. Layers of 1-qubit gates

Focusing first on the 1–qubit gates, the layer in equation (4) is composed of a sublayer,

$$\begin{equation} V_X\left(\theta\right) = \prod_{i = 1}^n R_{X_i}\left(\theta\right), \end{equation} \tag{ 10 }$$

of X rotations with the same angle θ. Since $\pi V_X(\theta) \pi^{-1} = V_X(\theta)$ for any permutation $\pi \in S_n$, we have $V_X(\theta) \in \mathcal{E}({\mathrm{Aut(H)}})$ for any H. However, this comes at the expense of imposing the same rotation to be applied to each of the qubits, and seems poorly tailored to the exact details of H.

A key insight is to realize that while it is reasonable to ask that equivalent qubits, with respect to H, should experience similar transformations, this does not have to extend to all qubits. To formalize further this notion of equivalence we introduce the concepts of orbits. (Orbit of H).

Definition 3 An orbit of a Hamiltonian H acting on a system of n qubits is defined as a set of qubit indices closed under the action of Aut(H). The orbit of an arbitrary qubit–index i is given by

$$\begin{equation} \mathcal{O}_{i} = \left\{\pi_a\left(i\right)\right\}_{\pi_a \in {\mathrm{Aut}\left(H\right)}}\,. \end{equation} \tag{ 11 }$$

When $j\in \mathcal{O}_{i}$, both the orbits $\mathcal{O}_i$ and $\mathcal{O}_j$ contain the exact same indices, and are thus deemed equivalent. We label non-equivalent orbits as $\mathcal{O}_o$ with a separate index o taking values in a subset of $\{1,\ldots, n\}$. Also, we denote the set of all distinct orbits $\mathcal{O} = \{\mathcal{O}_o\}$, which has cardinatlity $|\mathcal{O}|$ and forms a partition of the set of all qubit indices. Construction of orbits are illustrated in figure 2 for a system admitting 2 reflection symmetries, resulting in $|\mathcal{O}| = 3$ orbits numbered as 1, 2 and 3 in figure 2(b).

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The ORB generalization of $V_X(\theta)$ in equation (10) is defined as:

$$\begin{equation} V_X^\mathrm{\,ORB} \left(\boldsymbol{\theta}\right) = \prod^{|\mathcal{O}|}_{o = 1} \prod_{i \in \mathcal{O}_o} R_{X_i} \left(\boldsymbol{\theta}_o\right), \end{equation} \tag{ 12 }$$

which applies the same rotation to any qubit within the same orbit, but allows for different rotations in between distinct orbits. One can readily verify that any further division of this parameter grouping (i.e. the addition of any extra free parameter) would break the equivariance of the layer. That is, the parameter structure in equation (12) maximizes the number of independent parameters while still maintaining equivariance of the layer.

In the case when $\mathrm{Aut}(H) = \{I\}$, each qubit belongs to its own orbit so that each rotation is assigned a distinct angle, yielding $|\mathcal{O}| = n$ parameters per layer. When $\mathrm{Aut}(H) = S_n$, each qubit belongs to the same orbit and $|\mathcal{O}| = 1$. In general, $V^\mathrm{\,ORB}_X$ packs a number of parameters $|\mathcal{O}|$ in between 1 and n, that is the maximum possible (for a given H) while still preserving equivariance. While exemplified here for X-rotations, this construction equally applies to any layer of 1-qubit rotations.

3.3.2. Layers of 2-qubit gates

The concept of orbits can be extended to orbits over pairs of qubits (called edges). (Edge-orbit of H).

Definition 4 An edge-orbit of a Hamiltonian H acting on a system of n qubits is defined as a set of edges that remain closed under the action of Aut(H). The orbit generated by an arbitrary edge (i, j) is given by

$$\begin{equation} \mathcal{O}^{e}_{\left(i,j\right)} = \left\{\left(\pi_a\left(i\right),\pi_a\left(j\right)\right)\right\}_{\pi_a \in {\mathrm{Aut(H)}}}\,. \end{equation} \tag{ 13 }$$

As for orbits, any two edge-orbits are either exactly the same or non-intersecting, and the set of distinct edge-orbits $\mathcal{O}^{e} = \{\mathcal{O}^{e}_{m}\}$ partitions the full set of edges. We can further specialize this definition to edges in a subset $\mathcal{I}$ of all possible edges corresponding to the physical interactions of the system, that is, which appear as two–body terms in H. It can be verified that such restriction is consistent, i.e. given an edge (i, j) appearing in H any of its transform $(\pi_a(i), \pi_a(j))$ for $\pi_a \in {\mathrm{Aut}(H)}$ will also appear in H (by definition of ${\mathrm{Aut}(H)}$). Construction of edge-orbits are illustrated in figure 2(c) for the example mentioned earlier.

As before, we can generalize layers of 2–qubit gates appearing in HVA-like circuits by grouping parameters per edge-orbits. For instance, the layer of 2-qubit rotations $V_{ZZ}(\theta) = \prod_{(i,j) \in \mathcal{I}} R_{Z_i Z_j}(\theta)$ appearing in equation (4) becomes:

$$\begin{equation} V_{ZZ}^\mathrm{\,ORB} \left(\boldsymbol{\theta}\right) = \prod^{|\mathcal{O}^e|}_{m = 1} \prod_{\left(i,j\right) \in \mathcal{O}^e_m} R_{Z_iZ_j} \left(\boldsymbol{\theta}_m\right), \end{equation} \tag{ 14 }$$

that could pack up to $n(n-1)/2$ distinct parameters, as opposed to a single parameter for HVA-like circuits, while still retaining equivariance under ${\mathrm{Aut}(H)}$. As was the case for equation (12), one can verify that any additional independent parameter would break equivariance of equation (14).

This equivariant construction can be extended to any set of 2–qubit (and in principle to higher order) interactions as

$$\begin{equation} V_{P}^\mathrm{\,ORB} \left(\boldsymbol{\theta}\right) = \prod^{|\mathcal{O}^e|}_{m = 1} \text{exp}\left[- i \boldsymbol{\theta}_m \sum_{\left(i,j\right) \in \mathcal{O}^e_m} P_{i,j}\right], \end{equation} \tag{ 15 }$$

where P denote the type of interactions involved in between pairs of qubits.

When any of the $P_{i,j}$ interactions intra-orbit commute, equation (15) is readily implemented as a composition of 2-qubit gates $R_{P_{i,j}}(\boldsymbol{\theta}_m) = \text{exp}[-i \boldsymbol{\theta}_m P_{i,j}]$. This was the case for P = ZZ, resulting in equation (14). However, cases where interaction terms do not commute within orbits require more care, and will be detailed later in section 4.5. A simple example of application of ORB circuits with layers of the form (equations (12) and (14)) is illustrated in figure 3 and discussed in the next section.

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In the following, we aim at studying the performances of ORB circuits—especially their scaling with respect to increased problem sizes—and at comparing them to other related ansatzes. We now review how these different aspects are probed.

The performance of a given circuit, at fixed number of layers L, can be assessed in terms of the energy $E^\mathrm{opt}(L)$ achieved after parameter optimization, relative to the true ground-state energy $E_\mathrm{GS}$ of the Hamiltonian under study. This is quantified in terms of the relative error

$$\begin{equation} r\left(L\right) = \frac{E^\mathrm{opt}\left(L\right)-E_\mathrm{GS}}{|E_\mathrm{GS}|}. \end{equation} \tag{ 16 }$$

To gauge the performance of a circuit construction across different system sizes, we will report the critical number $L_c(\varepsilon)$ of layers required to achieve a sufficiently small error ε. For each family of circuits and each system size n considered, this number is obtained by repeating optimizations of circuits with increasing number of layers L until $r(L) \unicode{x2A7D} \varepsilon$. Additionally, we will report the total number $N_c(\varepsilon) = n_l \times L_c(\varepsilon)$ of independent parameters corresponding to $L_c(\varepsilon)$ layers and where n_l denotes the number of independent parameters per layer (that varies depending on the family of circuits considered).

All the circuits are numerically simulated using Tensorflow Quantum [43] and their optimizations are performed using the L-BFGS-B routine [44] with initial random circuit parameters and a number of iterations systematically capped to 500 iterations per individual optimization. Given such stochastic initialization, any optimization is repeated $N_\mathrm{rep} = 25$ times and results are reported in terms of the statistics (median) evaluated over these repetitions.

These optimization metrics already capture some main aspects of trainability. Namely, non-vanishing relative errors arise from limited expressibility of the circuits, and also from features of the optimization landscape such as the presence of low-quality local minima. Still, given that these figures are obtained on simulations, they do not reflect problems of vanishing gradients. Hence, we will also provide an assessment of the scaling of the variance of the parameter gradients. Variances $\text{Var} (\partial_i E)$ of the individual partial derivatives (see equation (5)) are evaluated over repeated random circuit initializations, and depend both on a given parameter index i and on the depth L of the circuit considered. To provide synthetic but meaningful information, we will report the median

$$\begin{equation} \text{Var}_{c}\left(\varepsilon\right) \equiv \langle \text{Var}\left(\partial_i E\right)\rangle_i \text{, for } L = L_c\left(\varepsilon\right) \end{equation} \tag{ 17 }$$

evaluated at a critical number of layers, that is, in a regime where high-fidelity ground state preparation is in principle possible.

As a starting example, we consider the TFIM with Hamiltonian

$$\begin{equation} H_{\text{TFIM}} = - \sum_{\langle i, j\rangle} Z_{i}Z_{j} - h_x \sum^n_{i = 1} X_i \end{equation} \tag{ 18 }$$

consisting of $Z_{i} Z_{j}$ interactions between neighboring pairs of qubits $\langle i,j \rangle$, and a global transverse field $\sum_i X_i$ affecting equally each of the n qubits with strength $h_x\gt0$.

In 1D, qubits are arranged over a line such that a qubit $i \in [2, n-1]$ has neighbors $i\pm1$. Additionally, for periodic boundary condition (i.e. a ring) qubits 1 and n are taken to be neighbors, while for open-boundary conditions (i.e. a path) both qubits 1 and n have a single neighbor. This 1D model is known to exhibit a transition from a ferromagnetic phase ($h_x\lt1$) to a disordered paramagnetic one ($h_x\gt1$), and can be exactly solved by means of the Jordan–Wigner transformation. Such minimal model of a quantum phase transition has been extensively used as a prototype of quantum magnetism and as a testbed for the assessment of VQA performances; it will serve us as a first benchmark.

In [11, 12], high–fidelity preparation of the ground state of the TFIM on a ring is consistently achieved using HVA circuits with critical depth and number of parameters growing linearly with the system size. The circuits employed have layers of the form equation (4) applied to an initial state $\vert \psi_0\rangle = \vert +\rangle^{\otimes n}$. In this case, the Hamiltonian remains invariant under any translation $\mathcal{T}_k(i) = i+k\, (\text{mod } n)$ of the qubits by a number of sites k. Given the 1D nature of the system, any qubit j can be mapped to any other qubit jʹ with $\mathcal{T}_{j^{^{\prime}} - j}$. In other words, each qubit (and edge) is equivalent to another, thus yielding a single orbit (and a single edge–orbit) as per definitions 3 and 4; in this case ORB circuit effectively recovers the HVA ansatz.

The situation differs when studying the same system with open boundary conditions, which is depicted in figure 3(a). Now, the Hamiltonian only admits a single non-trivial automorphism $\mathcal{R}(i) = n+1-i$, that permutes qubits from each side of the chain. Accordingly, qubits group as $|\mathcal{O}| = n/2$ ($|\mathcal{O}| = (n+1)/2$) orbits, for even (odd) values of n. Each orbit consists in either a pair of qubit indices $(i,n+1-i)$ or a single one when $i = (n+1)/2$. Similarly, edges groups as $|\mathcal{O}^e| = n/2$ ($|\mathcal{O}^e| = (n-1)/2$) edge-orbits for even (odd) n. Consequently, an ORB circuit allows for $n_l = n$ free parameters per layer, a figure to be compared to the $n_l = 2$ parameters for the HVA.

Both a layer of HVA and an ORB circuit are depicted in figure 3(b) for a system size of n = 5 sites. For each ansatz, independent parameters are displayed as distinct colors. For benchmarking purposes, we also include a 'Free' circuit, that consists of the same set of gates as the HVA and ORB circuits, but with each gate being assigned an independent parameter yielding a total of $n_l = 2n-1$ parameters per layer.

Naively, one would have expected the expressibility of the circuits to increase monotonically with the number n_l of free parameters per layer. However, it can be shown [34] that both ORB circuit and HVA have the same expressibility, growing quadratically with the system-size. In other words, despite the possibility of packing $n/2$ times more parameters per layer, ORB circuit spans (in the limit of large n) exactly the same group of unitaries as its HVA alter-ego. In contrast, the 'Free' ansatz has increased expressibility [34], and overall we have $\text{dim}(\mathfrak{g}_{Free}) \gt \text{dim}(\mathfrak{g}_{ORB}) = \text{dim}(\mathfrak{g}_{HVA}) = n^2$. We now probe further how such appealing feature translates into improved performances.

Figure 3(c) reports the critical number $L_c(\varepsilon)$ of layers (left panel), and critical number $N_c(\varepsilon)$ of parameters (right panel), required to achieve errors as low as $\varepsilon = 10^{-5}$. As can be seen, both ORB circuit and the 'Free' construction requires a number of layers substantially smaller than HVA. Already for system sizes of n = 15 qubits this corresponds to more than an order of magnitude reduction in the number of layers required, a significant edge in light of current noise levels of quantum hardware (e.g. see [15]).

Furthermore, ORB circuit is found to necessitate the least number of parameters to achieve a similar performance when compared to the other circuits studied. Note that the observed quadratic scaling in the number of parameters matches closely the dimension of the DLAs consistently with [32]. Overall, in this example, ORB circuit emerges as an optimal construction, in terms of number layers and parameters required, allowing us to pack as many parameters per layer as is possible without increasing the expressibility of the ansatz.

While 1D problems already offer invaluable insights, the most compelling, and challenging, physical models often arise in higher dimensions. We now study further applications of ORB circuit to the 2D TFIM laid over regular grids composed of $n_r \in [2,4]$ rows and $n_c \in [2, 7]$ columns involving up to n = 21 qubits (depicted in figure 3(a)).

The ansatzes studied earlier directly extend to this 2D scenario. The sublayer of interactions $V_{ZZ}(\theta) = \sum_{\langle i,j\rangle} R_{Z_iZ_j}(\theta)$, appearing in an HVA ansatz, now runs over all the pairs of adjacent grid qubits $\langle i,j \rangle$, but is still parameterized by a single parameter. Additionally, X rotations are applied to each of the n qubits with the same angle per layer. In the case of ORB circuits these X (ZZ) rotations are grouped by orbits (edge-orbits) resulting from the two reflections (one horizontal and one vertical) and also, in the case of $n_r = n_c$, from two additional ones (w.r.t. the diagonals). In the case of the Free ansatz, each gate corresponds to a distinct parameter.

Performances of the circuits for all the system sizes considered are depicted in figure 4(b) including the critical number of layers (top row), number of parameters (middle row) and gradients variances (bottom row) for a relative ground-state error of $\varepsilon = 10^{-3}$. In line with previous results, we find ORB circuit to exhibit enhanced performances. For the range of layers considered ($L\unicode{x2A7D} 500$), HVA circuits fail in converging for problems involving more than 10 qubits. In contrast both the Free and ORB circuit achieve the desired accuracy in $L_c(\varepsilon)\unicode{x2A7D} 12$ layers for all the system sizes probed. Furthermore, ORB circuits necessitate a number of parameters significantly smaller than the Free ansatzes, with 4 to 5 times less parameters for the largest system studied.

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We also compare the variance of the circuit gradients, defined in equation (17), for the three ansatzes. Except for the smallest sizes studied, the gradients of ORB circuits are found to be systematically larger than others. On one hand, HVA and ORB circuits share the same expressibility [34], such that at similar circuit depth one would expect similar gradient magnitudes. However, for a given error ε, ORB circuits require significantly less layers than HVA such that its gradients remain larger. On the other hand, both ORB and Free circuits converge in a similar number of layers, but the expressibility of Free circuits [34] is the largest, hence yield smaller gradients. Overall, these combined effects result in gradients of ORB circuits that are one or two orders of magnitude larger than others. (This discussion does not account for noise-induced gradient suppression [15]. Accounting for noise would lead to an even more dramatic separation between ORB and HVA, since HVA requires much deeper circuits than ORB.)

Such substantial improvement offered by ORB circuits should not obscure the fact that the dimension of the DLAs of any of the circuits grows exponentially as a function of n [34], hinting to issues of trainablity for too large system sizes [9]. To probe further the scaling of the circuits properties, we focus on the case of the $2\times n_c$ grids, and report in figure 5 the critical number of layers (top row) and gradients variances (bottom row) as a function of the system size $n = 2\times n_c$. As can be seen, the number of layers required to achieve precise ground-state preparation displays an exponential behavior over most of the system sizes studied, albeit at a much slower rate for ORB and Free circuits. Concurrently, magnitudes of the gradients decay exponentially fast over these system sizes. Still, we note a flattening of these exponential increases (in the number of layers) and decays (in the gradients magnitudes) for $n\unicode{x2A7E}16$ especially for ORB circuits. Notably, for n > 10 qubits, the gradients of ORB circuits are found to be significantly larger than for the HVA and Free circuits. That is, while not generically solving problems of BPs at large n, ORB circuits are found to mitigate them.

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Having demonstrated the benefits of ORB circuit for physical models defined on regular lattices, we now aim at understanding its appeal when dealing with less-to-unstructured Hamiltonian topologies. This situation is best exemplified for problems of Max-Cut addressed by means of the QAOA [2].

4.4.1. QAOA for Max-Cut

Recall that for a given graph $\mathcal{G}$, with set of n vertices $\mathcal{V}(\mathcal{G})$ and edges $\mathcal{E}(\mathcal{G})$, a Max-Cut Hamiltonian is defined (up to rescaling and constant shift) as

$$\begin{equation} H_{\text{MC}}\left(\mathcal{G}\right) = \sum_{\langle i,j \rangle \in \mathcal{E}\left(\mathcal{G}\right)} Z_i Z_j\, , \end{equation} \tag{ 19 }$$

that is, similar to equation (18) for $h_X = 0$, but, with interactions now depending on a potentially unstructured $\mathcal{G}$.

The ground state of equation (19) can be interpreted as the Max-Cut of $\mathcal{G}$ (i.e. the bipartition of the set $\mathcal{V}(\mathcal{G})$ that maximizes the number of edges connecting each part). As such, Max-Cut problems can be recasted as tasks of ground-state preparations, that are typically performed using QAOA circuits. Similar to the HVA for the TFIM model, a layer of QAOA is composed of collective ZZ rotations (now defined in correspondence to the set of edges $\mathcal{E}(\mathcal{G})$) and collective X rotations, accounting for $n_l = 2$ parameters per layer.

4.4.2. Vanishing symmetries in random graphs

Of importance to understand the following results, we recall that random graphs become almost surely asymmetric as their sizes increase. In other words, for large enough system sizes, a typical random graph admits no non-trivial automorphisms (see [45] and references thereof, for a more precise statement and assumptions). In the context of constructing ORB circuits for problems of Max-Cut, this implies that each node (edge) become structurally independent of any other node (edge), such that independent parameters are assigned to any of the parameterized gates composing the circuits. Hence, as the size of the underlying graphs are increased ORB circuits would become more and more similar to Free circuits.

This aspect is illustrated in figure 6 for 3–regular graphs spanning system sizes n = 4 to 20 nodes. For n = 4 nodes, the only 3-regular graph is a complete graph, and due to the many symmetries present all the qubits (pairs of qubits) group into a single orbit (edge-orbit). In this case the QAOA and ORB circuit are the same. When the size of the random graphs is increased, the number of symmetries quickly decrease, and the qubits (pairs of qubits) group into increasingly more distinct orbits (edge-orbits). Already, for n = 10 (16) nodes, this yields a total of $n_l = 13$ (37) parameters. This number still remains smaller than the number of parameters for the Free ansatz which is given by $n_l = |\mathcal{V}(\mathcal{G})|+|\mathcal{E}(\mathcal{G})|$ (i.e. a single parameter per node and per edge of $\mathcal{G}$), and which for 3-regular graphs equals $n+3n/2 = 25$ (40). Finally, the graph with n = 20 nodes only admits the identity as a trivial symmetry, such that each gate in ORB circuit has its own parameter (i.e. in this case ORB and Free circuits become effectively the same), in stark contrast with the $n_l = 2$ parameters per layer of QAOA. Overall, we see that ORB can recover both QAOA or Free circuits depending on the amount of symmetries present in the underlying graph, but in general will be different.

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4.4.3. Results

The performances of the three circuit constructions are reported in figure 7 for $\varepsilon = 10^{-3}$. As seen in figure 7 (top panel), the critical number of layers for QAOA circuits is found to grow exponentially with n, and for n > 10 nodes we could not achieve convergence given a maximum number of 1000 layers. Still, both the ORB and Free circuits consistently converge within $\mathrm{ten}$ layers, with a small difference between their performances for $n \unicode{x2A7D} 10$ nodes. Such behavior is to be expected in light of the previous comments. As the size of the random graphs increases, performances of the Free and ORB circuits become increasingly similar. Remarkably, when looking at the critical number of parameters required (mid panel) and magnitude of the initial gradients (bottom panel), it can be seen that ORB circuits systematically overperform other circuit constructions, gracefully interpolating between the best features of QAOA and Free circuits.

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Overall, the performance and trainability of ORB circuits are found to be substantially better than those entailed by alternative circuit constructions. Again, we stress that, for the cases we studied, gradients are found to decrease exponentially fast with the system size, but, at a slower pace for ORB circuit .

4.4.4. Discussion

Going beyond models of matter involving only local interactions or commuting terms, ORB circuit readily extends to more challenging physical problems. To showcase these possibilities, and address the practical issues arising in such cases, we conclude this section by studying the ground state preparation of a frustrated J₁–J₂ Heisenberg model [47] laid on a grid.

The Hamiltonian of the system, for a $3\times4$ grid depicted in figure 8(a), is defined as

$$\begin{align} H_{J_1\text{-}J_2} = J_1 \sum_{\langle i,j \rangle} S_{ij} + J_2 \sum_{\langle \langle i,j \rangle \rangle} S_{ij}, \end{align} \tag{ 20 }$$

in terms of the Heisenberg exchange interactions $S_{ij} = X_iX_j + Y_iY_j + Z_iZ_j$ between pairs $\langle i,j \rangle$ of nearest-neighbor spins and pairs $\langle \langle i,j \rangle \rangle$ of next–nearest–neighbors, with strength J₁ and J₂ respectively. Due to these competing interactions, this model reveals a rich physics and remains subject of speculation w.r.t. the existence of a spin liquid phase, and potentially a second phase, in the vicinity of $J_2/J_1 = 0.5$ [48]. As such, it has been used as a testbed for the study of state-of-the-art numerical methods and the development of novel circuit constructions [25, 49–51]. For example, in [25] an ansatz was proposed respecting the $\mathbb{SU}(2)$ invariance of equation (20) for the purpose of ground state preparation that will serve us as a benchmark.

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In the following, we assume access to a quantum device with 2D neighbors-coupling (i.e. allowing for 2-qubit interactions between neighboring grid qubits). Given the $\mathbb{SU}(2)$ symmetry of equation (20), we employ gates of the form $R_{S_{i,j}} = \exp [-i \theta S_{i,j}]$, over pairs of neighbors $\langle i,j \rangle$, that commute with the action of the symmetry group [19, 25]. We stress that the set of gates employed here is only a subset of the terms appearing in equation (20), but we do not require extended next-nearest-neighbor connectivity.

The set of edge-orbits, restricted to pairs of neighbor qubits, is readily obtained by considering the two spatial symmetries of the grid (a horizontal and a vertical reflection). This yields a total of 6 edge-orbits labeled in figure 8(b) from a to f. While some of the edge-orbits only involve non intersecting edges (a, b, e, and f), this is not always the case (c and d). In practical terms, this means that due to the non-commuting nature of gates $R_{S_{i,j}} R_{S_{i,k}}$ sharing a common qubit i, care needs to be taken when implementing ORB layers of the form equation (15) under the assumption of 2-qubit connectivity.

Consider as an example the two edges (1,5) and (5,9) belonging to the orbit c. Equation (15) would require to realize the unitary $R(\theta) = \exp[-i\theta (S_{1,5} + S_{5,9})]$, which cannot be directly implemented as a composition of $R_{S_{1,5}}(\theta)$ and $R_{S_{5,9}}(\theta)$. Still, two options are possible. (i) One can compile $R(\theta)$ as gates $R_{S_{1,5}}$ and $R_{S_{5,9}}$ with suitable angles. Given that the Lie algebra generated by the terms $S_{1,5}$ and $S_{5,9}$ has dimension 4, compilation of $R(\theta)$ would require a total of at least 4 gates. (ii) As an alternative which does not increase the physical depth of a layer, one can slightly relax the equivariance requirement. This is achieved by decorrelating parameters associated to overlapping gates belonging to a same edge-orbit, and results in our case in 2 additional parameters, one for each of the edge-orbit c and d. Given our aim for shallow circuit constructions, we follow the second option. Overall, the ORB circuit involves a total of 4 rounds of gates that can be performed in parallel (we call such a round a sublayer), with a total of $n_l = 8$ parameters per layer, depicted in figure 8(c).

In comparison, a layer of HVA for such problem (extending the construction for the 1D Heisenberg chain in [11]) would also necessitate 4 sublayers—this is the minimum number in order to realize a full set of R_S gates over all the edges of the grid—and $n_l = 4$ parameters. Such ansatz corresponds to the sublayers displayed in figure 8(c) but with now a single parameter per sublayer (i–iv). The Free ansatz consists in the same sublayers with 1 parameter per gate and a total of $n_l = 17$ parameters per layer.

As studied in [11, 12, 25], the circuits are initialized as a tensor product of singlet states $\vert s\rangle = (\vert 01\rangle-\vert 10\rangle)/\sqrt{2}$, which are defined here over adjacent qubits composing each row of the underlying grid. Such a state has zero total spin and magnetization values that remain conserved during its evolution through the quantum circuit due to the choice of the R_S gate-set. Furthermore, it can be verified that this initial state is invariant under the spatial symmetries of the grid.

For the frustrated case with $J_1 = 1$ and $J_2 = 0.5$ studied in [25], we achieve a relative ground state error $\varepsilon\lt5\times10^{-3}$ using ORB circuit with as few as L = 5 layers, 85 gates and 40 parameters, a significant improvement compared to the circuit presented in [25] that requires several hundreds of gates, most of them involving non-neighbor qubits (i.e. that would necessitate further extensive decomposition), and as many parameters. Furthermore, and consistent with previous results, ORB circuit is found to perform better than HVA, which requires $L_c(\varepsilon) = 30$ layers. It also performs well compared to the Free ansatz, which requires a similar number $L_c(\varepsilon) = 4$ of layers, but twice as many parameters, and exhibits five times smaller initial gradients.

As was discussed in section 4.4, in the context of Max-Cut problems when the underlying graphs do not have any symmetries (as would most typically happen for large random graph) ORB circuits naturally recover the ma-QAOA circuits [46]. However, when a few symmetries persist, ORB-circuits differ from ma-QAOA and we saw numerically its advantage. Furthermore, studies performed in this work were conducted in a different regime, with relatively large number of layers exceeding the numbers $L = 1-3$ considered in [46].

References [23, 51] restore symmetries of a state prepared by a quantum circuit, which does not preserve symmetries of the problem, by means of additional measurements and post-processing, as opposed to the direct circuit construction presented here.

In [64, 65] the authors considered spatial symmetries and algebraic tools that are similar to the ones underpinning this work, but, limit their usage to performance predictions [64] or to the acceleration of numerical simulations of certain types of low–depth quantum circuits [65].

Finally, [20] proposes designing equivariant circuits through a process of gate symmetrization. We highlight that this symmetrization strategy can, when specialized to spatial symmetries, recover the construction proposed. In [20], the merit of such symmetrization is assessed in the context of QML tasks, and also for fixed system-size problems of ground state preparations. Here, the benefit of incorporating spatial symmetries is demonstrated as a scaling effect, and its use is further theorized from an circuit expressibility perspective.