SU(d)-symmetric random unitaries: quantum scrambling, error correction, and machine learning

Abstract

Quantum information processing in the presence of continuous symmetry is of wide importance and exhibits many novel physical and mathematical phenomena. SU(d) is a continuous symmetry group of particular interest since it represents a fundamental type of non-Abelian symmetry and also plays a vital role in quantum computation. Here, we explicate three particularly interesting applications of SU(d)-symmetric random unitaries in diverse contexts ranging from physics to quantum computing: information scrambling with non-Abelian conserved quantities, covariant quantum error correcting random codes, and geometric quantum machine learning. First, we show that, in the presence of SU(d) symmetry, the local conserved quantities would exhibit residual values even at t → ∞ which decays as Ω(1/n^3/2) under local Pauli basis for qubits and $\Omega (1/{n}^{{(d+2)}^{2}/2})$ under local symmetric basis for general qudits with respect to the system size, in contrast to O(1/n) decay for U(1) case and the exponential decay for no-symmetry case in the sense of out-of-time ordered correlator (OTOC). Second, we show that SU(d)-symmetric unitaries can be used to construct asymptotically optimal (in the sense of saturating the fundamental limits on the code error, or the approximate Eastin–Knill theorems) SU(d)-covariant codes (codes with universal transversal gates) that encode any constant number of logical qudits, extending [Kong & Liu; PRXQ 3, 020314 (2022)]. Finally, we derive an overpartameterization threshold via the quantum neural tangent kernel (QNTK) required for exponential convergence guarantee of generic ansatz for geometric quantum machine learning, which reveals that the number of parameters required scales only with the dimension of desired subspaces rather than that of the entire Hilbert space. Our work invites further research on quantum information with continuous symmetries, where the mathematical tools developed in this work are expected to be useful.

Introduction

Symmetries serve as not only fundamental principles in theoretical physics, but also vital roles in modern quantum technologies. Specifically, within the vast array of quantum circuit models, the adoption of common symmetries such as rotations or permutations under group actions unveils innovative features across various facets of contemporary quantum information science and quantum computing. A principal emergence of symmetry in quantum information is to study random circuits with conservation laws, which has recently drawn significant attraction due to its fundamental relevance in theoretical physics and rich phenomena displayed therein, including quantum information scrambling^{1,2,3,4,5,6,7,8,9,10}, quantum error correction^{11,12,13,14,15}, quantum machine learning^{16,17,18,19,20,21,22}, and random benchmarking protocols^23,24,25.

The famous Noether’s theorem posits that in a closed quantum system, continuous symmetries are associated with conserved charges, namely, conservation laws. The entire Hilbert space is subsequently decomposed into charge sectors whose charges are reflected through the eigenvalues of the underlying conserved charge or the Casimir operators. Quantum systems governed by conservation laws play a key role in quantum information theory, exemplified by foundational results such as the Eastin–Knill theorem for quantum error-correcting codes^26,27,28. Recent work has uncovered further constraints imposed by conservation laws, including limitations on information recovery^{10,11,29,30,31} and no-go theorems to achieve universal quantum computation via local interactions^18,32,33,34. Among these, random quantum circuits with conservation law produce many novel physical insights such as in operator spreading^{35,36,37,38,39,40}, covariant quantum error correction^14,15, and monitored circuit dynamics^41,42.

In this paper, we are specifically interested in the SU(d) symmetry, a canonical model of non-Abelian symmetry governed by the special unitary group SU(d) acting transversally on each qudit with local dimension d over the system and tightly correlated with the permutation symmetry of the system, which widely exists in quantum many-body systems, quantum chemistry, and graph-based data structures, due to a mathematical formalism called Schur–Weyl duality^43,44. The theory of random quantum circuits with SU(d) symmetry has recently produced fruitful implications e.g., see refs. ^15,18,41,45 and drawn interest due to non-Abelian conserved charges. Here, we primarily focus on three different applications: quantum scrambling, quantum error correction, and quantum machine learning.

Conserved quantities from SU(d)-symmetric quantum circuits, due to Schur–Weyl duality, are generated by the symmetric group S_n which permutes the sites of qudits. Except for the aforementioned significance, SU(d)-conserved quantities serve as a natural and important generalization to that of U(1) symmetry which has been systematically studied in terms of hydrodynamics. To be specific, it has been observed that the diffusion of local U(1) conserved charges scale in power law in time and in late time post scrambling, the existence of hydrodynamic tails or slow-mode relaxation^35,37. Moreover, in a generic chaotic system with energy conservation satisfying the eigenstate thermalization hypothesis (ETH), a similar late-time slow-mode relaxation exists which scales as power-law decay with system size⁴⁶ as opposed to the exponentially fast decay proved in the case without any conserved quantity^1,11,31,47. It is, hence, of interest to study the case with SU(d)-conserved quantities in the late time regime and check if the non-Abelian nature could further contribute to the slow-mode relaxation by probing out-of time-ordered correlator (OTOC)^48,49,50 which is widely used in qualifying quantum scrambling dynamics by measuring the spread of local charges under the dynamics in Heisenberg picture. As the first result of this paper, we mathematically derive a power inverse lower bound Ω(1/n^3/2) for the finite-size residual value of the OTOC due to the spreading of local conserved quantities with respect to SU(2) charge density (which is equivalent to using the local Pauli basis to probe the quantity). We further show that the finite-size residual value between two local conserved quantities via OTOC is $\Omega (1/{n}^{{(d+2)}^{2}/2})$ for SU(d)-symmetry on general qudits.

In the case where the symmetry acts transversally on local degrees of freedom, the symmetry actions can be understood as transversal implementations of logical gates in the context of quantum error correction. Codes that respect such symmetries are called covariant codes. For continuous symmetries like SU(d) and U(1), the Eastin–Knill theorem prevents the existence of covariant codes admitting perfect recovery without errors²⁶. Nevertheless, it is still possible to find approximate recovery schemes with approximate Eastin–Knill theorem^{14,27,28,51,52}. It claims a fundamental limit in terms of infidelity which asymptotically scales as O(1/n) where n is the number of physical qudits. One question of great interest is the existence of optimal quantum error correction codes that would saturate this fundamental limit with the presence of symmetry. We improve upon the previous construction of optimal SU(d)-covariant random codes¹⁵ to encode arbitrary k logical qudits. We prove that for k which does not scale with the number n of physical qudits, the code is asymptotically optimal to the fundamental limit against one-qudit erasure error. However, we provide numerical evidence that the code ceases to be optimal if we assume an overlarge coding rate against one-qudit erasure error.

Moreover, random circuits are currently applied to the study of the convergence rate of the near-term variational algorithms through quantum neural tangent kernel (QNTK)^20,21. In variational quantum algorithms and quantum machine learning, QNTK indicates the theoretical predictions of the convergence efficiency during the gradient descent process for given variational circuits. For a wide array of problems, especially for applications in physics, designing a variational ansatz that respects the underlying symmetry is crucial because it not only reflects the physical properties but is also observed to significantly improve the convergence speed and require much fewer parameters^18,53. It is speculated that the superior performance of symmetric-adaptive variational algorithms likely stemmed from the reduced effect of the barren plateau from its reduced effective dimension of the dynamical Lie algebra. For this reason, a recent subfield called geometric quantum machine learning has emerged^17,54, and it is of practical interest to study the convergence theory of near-time quantum variational algorithms that respect the underlying symmetry. To this end, we derive an overparametrization threshold for the existence of exponential convergence whose parameters scale with the dimension of a particular charge sector decomposed from the entire Hilbert space in which the information is encoded under SU(d) symmetry and, more generally, under any continuous symmetry generated by Lie algebra or discrete cases like permutation symmetry. This result generalizes many previous studies on the theoretical convergence on permutation invariant ansatz to general symmetries.

In this paper, we systematically investigate the above aspects in the presence of SU(d) symmetry. In Section II we provide necessary backgrounds regarding random unitary circuits and SU(d) symmetry. In Section III A we explain the power inverse lower bound of the OTOC under SU(d) symmetry. In Section III B we present the optimal SU(d)-covariant code encoding k qudits provided that k does not scale up with n. In Section III C, we calculate the overparametrization threshold for QNTK under general compact group symmetry and specify its applications to SU(d) and S_n symmetry. Our results may invite future research in several directions. On a fundamental level in quantum information, they motivate the study of the information scrambling from local interactions such as the local random circuits under non-Abelian conserved quantities. We anticipate that further work on entanglement generation and operator spreading with the non-Abelian conserved quantities would shed light on this phenomenon especially compared with the results in³⁶, and in the monitored random circuits under SU(d) symmetry. In addition, our work might connect with recent works on the fundamental limitation of recovering fidelity from the random circuits with a conservation law^10,27,29,30, especially through the lens of studying the mutual information (see in Eq. (6) in Section II). Furthermore, we expect that the results would find towards the ongoing effort in geometric quantum machine learning by providing insights on its convergence criterion and potential robustness to noise and barren plateau. All calculations and proof details as well as necessary mathematical backgrounds are pathologically presented in the Supplementary Material (SM).

Entanglement, decoupling, and OTOC under SU(d) symmetry

The above questions can be mathematically formulated in the same line through the concepts of OTOC and entanglement (mutual information). This connection is, perhaps, not surprising since all applications concern the questions of information detecting and recovery post scrambling. We borrow tools from refs. ^18,34 developed to study these concepts under SU(d) symmetry. In the case of SU(d) symmetry, the Hilbert space ${\mathcal{H}}$ is decomposed according to the irreducible representation (irrep) of the symmetric group as:

$${\mathcal{H}}=\bigoplus _{\lambda \vdash n}{1}_{{m}_{\lambda }}\otimes {S}^{\lambda },$$

(1)

where S^λ stands for a irrep with λ⊢n recording the irrep as a partition of the number n of qudits in the system^44,55. The number m_λ denotes the multiplicity of S^λ and $\dim {S}_{\lambda }\equiv {d}_{\lambda }$ is its dimension. A Haar SU(d)-symmetric random unitary $U{\equiv \bigoplus }_{\lambda \vdash n}{I}_{{m}_{\lambda }}\otimes {U}_{\lambda }$ with U_λ ∈ U(S^λ) is drawn from the compact group ${{\mathcal{H}}}_{\times }$ of SU(d)-symmetric unitaries under Haar measure, whose balance-kth-order expander (k-design) is:

$${T}_{k}^{{\rm{Haar}}}={\int}_{{{\!\!\mathcal{U}}}_{\times }}dU{U}^{\otimes k}\otimes {\bar{U}}^{\otimes k}.$$

(2)

Explicit expression of the above integral (presented in SM III.A) is at the center stage of most technical calculations in this paper for it connects well with OTOC and entanglement which we pay significant attention to. The relationship between OTOC and entanglement is explained gradually in the following context. As a reminder, even though they are tightly related physical quantities to probe the scrambling of quantum information, it is shown that separation of scales between these two quantities exists⁵⁶.

To begin with, suppose that Alice prepares a Bell pair $| {\phi }^{RA}\left.\right\rangle =\frac{1}{\sqrt{{d}_{A}}}{\sum }_{i}{\left\vert i\right\rangle }_{A}{\left\vert i\right\rangle }_{R}$ between the register R and A each endowed with k qudits of the system with d_A = d^k the dimension of A. Alice then transmits her information via the channel:

$${\Phi }^{RA}\mapsto U({\Phi }^{RA}\otimes \Psi ){U}^{\dagger }$$

(3)

which is a CPTP map given a proper density matrix Ψ (which could be purified by adding memory qudits). At some time Bob may measure part of the system of t qudits denoted by B (as well as the memory) so that Bob might in principle extract information from the state:

$${\rho }_{B\cup {\rm{MEM}}}=\,{{\rm{Tr}}}_{R\cup \bar{B}}U({\Phi }^{RA}\otimes \Psi ){U}^{\dagger }$$

(4)

A simple qualification of Bob’s detection on Alice’s information is to calculate the mutual information I(R: B ∪ MEM) = S(R) + S(B ∪ MEM) − S(R ∪ B ∪ MEM), and we say that Bob successfully detects all Alice’s information if I(R ∪ B ∪ MEM) saturates to its maximum value 2k. This scenario depicts the so-called Hayden-Preskill protocol¹, which has many implications in black hole physics and quantum gravity. With recent developments in black hole information, we can see the correspondence between a black hole and the setup in Fig. 1. To obtain the Page curve for the process of black hole radiation (working in the back ground of AdS/CFT), people isolated the radiation of the black hole from the CFT, which were usually called the reservoir⁵⁷ and then adding a bulk entanglement term to the Ryu-Takayanagi proposal^58,59. It should be clear that this corresponds to the state Ψ and MEM part in Fig. 1. Then a physical reason that we can choose Ψ to be pure is that we consider a black hole that were formed from a collapse of a pure state of matter consisting billions of atoms. The reservoir is entangled with the black hole interior which, at last, turns into the island. In Fig. 1, $\bar{B}$ can be considered as the island, since after the action of a random unitary, we should be looking at an old black hole. As a result, the information recovery discussion here has many physical implications in AdS/CFT correspondence^57,60, black hole information paradox^1,2,58,61, and quantum many-body teleportation⁶².

Fig. 1: Hayden-Preskill decoupling as a general set-up where R and A form a maximally entangled pair and Ψ is a generic density matrix in $\bar{A}$ which is purified with the memory register MEM.

In the case that Ψ itself is a pure state, we omit the MEM register for the case of SU(d)-covariant codes.

To lower-bound the mutual information, we may replace von Neumann entropy by Rényi entropy ${S}^{(2)}(\rho )=-\log {\rm{Tr}}({\rho }^{2})$ as a lower bound to appraise the information recovery scheme, which also enjoys a more convenient computation integrating over the concerned group. Assume ${\Phi }^{RA},\Psi ={\Psi }^{\bar{A}{\rm{MEM}}}$ are all given by Bell pairs, then

$${\rho }_{\bar{B}}=\,{{\rm{Tr}}}_{R\cup B\cup {\rm{MEM}}}U({\Phi }^{RA}\otimes \Psi ){U}^{\dagger }=\frac{{I}_{\bar{B}}}{{d}^{n-t}}$$

(5)

and one can check by Schmidt decomposition that ${S}^{(2)}(\bar{B})={S}^{(2)}(R\cup B\cup {\rm{MEM}})$. Since von Neumann entropy S(B ∪ MEM), in general, dominates S⁽²⁾(B ∪ MEM), I(R: B ∪ MEM)≥I⁽²⁾(R: B ∪ MEM) and we define the information recover fidelity under Haar average as (see computational details in SM III.A)

$$\begin{array}{rcl}&&{\mathbb{E}}\left[{S}^{(2)}(R)+{S}^{(2)}(B\cup MEM)-{S}^{(2)}(R\cup B\cup MEM)\right]\\ &\ge &2(n+k)-\log \left(\sum\limits_{{P}_{A}{P}_{B}}\int\,dU\,{\rm{Tr}}({\widetilde{P}}_{A}^{\dagger }{P}_{B}^{\dagger }{\widetilde{P}}_{A}{P}_{B})\right)\end{array}$$

(6)

where ${\widetilde{P}}_{A}\equiv U{P}_{A}{U}^{\dagger }$ for a given Haar random unitary and Pauli strings on A. We explicate in Eq. (13) in the following that this lower bound can be obtained computing a sum of OTOC with respect to the Pauli basis elements on the respective subsystems.

Alternatively, since $I(R:B\cup {\rm{MEM}})+I(R:\bar{B})=2k$ due to Schmidt decomposition, we could observe whether the mutual information between the register and Bob’s inaccessible region $\bar{B}$ is small. Then it is equivalent to lower bound the mutual information $I(R:\bar{B})=D({\rho }_{{\rm{R}}\cup \bar{{\rm{B}}}}\parallel {\sigma }_{R\cup \bar{B}})$ equal to the quantum relative entropy with respect to the product state

$${\sigma }_{R\cup \bar{B}}={\rho }_{R}\otimes {\rho }_{\bar{B}},$$

(7)

where ρ_R and ${\rho }_{\bar{B}}$ are the respective reduced density matrices on the subsystem R and $\bar{B}$, whose mutual information is zero in either von Neumann or Rényi sense. Therefore, mutual information is useful in probing any residual entanglements between R and $\bar{B}$: the smallness of $I(R:\bar{B})$ can be seen as a necessary condition for the decoupling given by the trace norm distance or purified distance between ${\rho }_{R\cup \bar{B}}$ and ${\rho }_{R}\otimes {\rho }_{\bar{B}}$, as a consequence of the continuity relation of the Fannes-Audenaert inequality (see SM I.C). This leads to the decoupling equation in Hayden-Preskill protocol^1,9, Page theorem⁶³ and the use of complimentary channel formalism in quantum error correction⁶⁴. Depending on the choice of norms such as the Schatten-p norm or purified distance, the computation would resort to applications of 2-designs and 1-designs. In particular, as we show in Section III B, we need to compute the following quantities:

$$\sum _{{P}_{A},{P}_{B}}\int\,dU\,{\rm{Tr}}\left(({P}_{A}^{\dagger }\otimes \Psi ){\widetilde{P}}_{B}^{\dagger }({P}_{A}\otimes \Psi ){\widetilde{P}}_{B}\right).$$

(8)

Simplification can be made if Ψ is further selected as a pure state (so there is no need for MEM states to purify) and then we obtain the following expression during the calculation process:

$$\frac{1}{{d}_{\bar{B}}^{2}-1}\frac{1}{{d}^{n}}\sum _{{P}_{\bar{B}}\ne I}\int\,dU\,{\rm{Tr}}(\widetilde{\Psi }{P}_{\bar{B}}^{\dagger }\widetilde{\Psi }{P}_{\bar{B}}).$$

(9)

The pure state $\Psi \equiv | \psi \left.\right\rangle \left\langle \right.\psi |$ defined on a sub-region $\bar{A}$ has no locality assumption. The above in mathematical form suggests similarity to another physical quantity: out-of-time-ordered commutator which might generally overestimate how fast the information propagates than that of entanglement^47,56,65. Given a chaotic Hamiltonian H the operator growth of a local observable under the Heisenberg picture W(t) ≡ e^iHtWe^−iHt is detected by the non-commutativity at certain observables at different sites initially. For the random circuit model, it is common to discretize the notion of time such as in the Brickwork model where each layer represents one time step. The non-commutativity is measured by the out-of-time-ordered commutator (OTOC):

$$\begin{array}{lll}{C}^{WV}(t)&=&\frac{1}{2}\,{\rm{Tr}}\left({[W(t),{V}_{r}]}^{\dagger }[W(t),{V}_{r}]\right)\\ &=&\;1-\,{\rm{Tr}}\left(W{(t)}^{\dagger }{V}_{r}^{\dagger }W(t){V}_{r}\right)\\ &=&\;1-{F}^{WV}(r,t),\end{array}$$

(10)

where V_r is a local traceless and normalized observable at site r such that $\,{\rm{Tr}}({V}_{r}^{\dagger }{V}_{r})$ equals one. At sufficiently large time post scrambling, we expect that the circuit would become Haar random or at least be unitary k-design (matching the k-th moment of the Haar randomness^66,67). Then the OTOC may be reformulated as ${F}^{WV}(r)={\lim }_{t\to \infty }{F}^{WV}(r,t)$ and we call it the finite-size residual value detected by the OTOC.

To obtain a more precise understanding, it is useful to think about the growth of the Heisenberg operator W(t) under a complete basis of operators, e.g., from the (generalized) Pauli group ${{\mathcal{P}}}_{n}={\{\langle {X}^{i}{Z}^{j};i,j = 0,...,d-1\rangle \}}^{\otimes n}/\langle \omega {{\bf{1}}}_{{d}^{n}}\rangle$ on qudits where $\omega =\exp (i2\pi k/d)$ for k = 0, … , d − 1, which obeys the following normalization and completeness properties for ${\mathcal{S}}\in {{\mathcal{P}}}_{n}$.

$$\begin{array}{l}\frac{1}{{\rm{tr}}\,I}{\rm{tr}}\left({{\mathcal{S}}}^{\dagger }{{\mathcal{S}}}^{{\prime} }\right)={\delta }_{{\mathcal{S}}{{\mathcal{S}}}^{{\prime} }},\\ \frac{1}{{\rm{tr}}\,I}\sum\limits_{{\mathcal{S}}}{{\mathcal{S}}}^{\dagger }\otimes {\mathcal{S}}=\Pi ,\end{array}$$

(11)

where Π is the SWAP operator between the first and second system, i.e., $\Pi | i,j\left.\right\rangle =| j,i\left.\right\rangle$ for i, j ∈ [dⁿ]. There are in total d²ⁿ different Pauli strings for n qudits. When d = 2 for qubits, this recovers the familiar Pauli matrices with identity σ^x, σ^y, σ^z, I. Then the Heisenberg operator W(t) can be expanded under the basis as

$$W(t)=\sum _{{\mathcal{S}}}\,{\rm{Tr}}({{\mathcal{S}}}^{\dagger }W(t)){\mathcal{S}}=\sum _{{\mathcal{S}}}\alpha ({\mathcal{S}},t){\mathcal{S}}.$$

(12)

Obviously V_r = I provides no nontrivial dynamics, we hence define the OTOC at late time by

$${F}^{W}(r)=\frac{1}{{d}^{n}}\frac{1}{{d}^{2}-1}\sum _{{V}_{r}\ne I}{\int}_{{{\!\!\mathcal{U}}}_{\times }}dU\,{\rm{Tr}}({\widetilde{W}}^{\dagger }{V}_{r}^{\dagger }\widetilde{W}{V}_{r}),$$

(13)

where V_r are taken to be nontrivial Pauli matrices supported on the site r with asymptotically Heisenberg operator ${\lim }_{t\to \infty }W(t)$ being replaced by its Haar random dynamics $\widetilde{W}=UW{U}^{\dagger }$ based on the assumption that we enter the scrambled regime that satisfies at least 2-design at late time. Up to coefficients, this definition uncovers the physical meaning of the lower bound of mutual information in our previous discussion. Without the presence of symmetry, the residual value F^WV(r) given by the OTOC is exponentially small in the system size n and independent of r as a consequence of non-locality^{11,47,49,50,65}. The intuition behind this exponential suppression of residual value in OTOC can be given in the following: $\widetilde{W}$ in the fully scrambled regime has support on all sites so that its expansion coefficients associated with the Pauli basis behave like Gaussian random variables from the law of large numbers. Hence, given V_r to be a single Pauli group element, half part of the expansion in Eq. (13) would anticommute and half commute so that they finally cancel. However, we present in Section III A that the late-time spreading of local SU(d) conserved quantities scale in an inverse power law with respect to the number n of sites in the system, which agrees with the generic observation of scrambling in the presence of conservation law^8,9,35,46.

To be noted, analytical computation to be above equations in the presence of SU(d) conservation law is highly non-trivial. Due to the generally super-polynomial scaling of the number of irreps λ⊢n, the number of basis elements spanning even 2-design commutant is intractable (see SM II.E for more details). To tackle the issue, we borrow techniques from ref. ^18,34 where we employ group representation-theoretical tools such as the Okounkov-Vershik approach⁶⁸. Additional information can be found in the self-contained Supplementary Material provided in this paper.

Results

Late-time saturation of SU(d) conserved quantities

The late-time hydrodynamics of other classes of chaotic systems with conservation laws has drawn significant interest in physics lately. It was shown⁴⁶ that for general energy-conserving quantum chaotic Hamiltonians assuming the eigenstate thermalization hypothesis (ETH), the OTOC for finite-size systems has a residual that scales as O(1/poly(n)) at late time. More precisely, the energy conservation in this setting is defined by imposing symmetry conditions on partitions of spectrum range by the energy difference Δ. The operator growth or transport of local charges under e.g. U(1) symmetry has received significant attention. It has been shown that the conservation law slows relaxation in OTOC, inducing a “hydrodynamic tail” at late times³⁵. In the post scrambling regime t → ∞, the locality is lost due to the assumption of Haar randomness (or at least being 2-design) under U(1) conservation. In this regime, it has been shown that the finite-size residual values between the local charges Z and the “raising charges” (defined by eigenmatrix of the adjoint action of the total charge Z_tot) scale O(1/n) (see also SM III.C for more details).

The SU(d) symmetry is generated by a set of non-commuting charges from the elements of the Lie algebra ${\mathfrak{su}}(d)$. The non-commuting nature implies that our charge sectors cannot exactly correspond to the spectrum of these conserved charges but rather its Casimir operators. The transversal action of the symmetry group SU(d) then implies there exists a sequence of coupled Casimir operators. For instance, in the case of SU(2) action, they are built by sequential coupling {S₁ ⋅ S₂, S₁ ⋅ S₂ ⋅ S₃, S₁ ⋅ S₂ ⋅ S₃ ⋯ S_n} of spin operators^69,70. Hence, the first sharp contrast between Abelian group symmetry and non-Abelian symmetry group is that the spectrum of conserved charges (or Lie algebra generators) in the latter case cannot explicitly correspond to charge sectors so that generally conserved quantities are not the same as conserved charges.

Here we define the conserved quantities to be elements generated by the exchange interactions S_i ⋅ S_j acting on the (i, j)-pair of qudits. We pay special attention to the dynamics of the 2-local conserved quantities given by the exchange interactions. In particular, we consider the OTOC between the charge density and exchange interactions under SU(2) symmetry action where the charge density is given by the non-identity single-qubit Pauli elements. We also consider the OTOC between exchange interactions for general transversal SU(d) symmetry. At the first and second qudits, it reads

$$W=\frac{1}{\sqrt{{d}^{2}-1}}{{\bf{S}}}_{1}\cdot {{\bf{S}}}_{2}=\frac{1}{\sqrt{{d}^{2}-1}}\sum _{P\ne I}P\otimes {P}^{\dagger }$$

(14)

where P are generalized Pauli matrices acting on one qudit. In the case of d = 2 we have S_i = (X_i, Y_i, Z_i). The first result of this paper is a rigorous mathematical verification that under SU(d) symmetry the late-time OTOCs of these local conserved quantities scale as an inverse polynomial in contrast to exponential decay when there is no symmetry. At late time we work with the Heisenberg operator $\widetilde{W}=UW{U}^{\dagger }$ where U is randomly taken from SU(d)-symmetric unitaries. As mentioned at the end of Section II, we take a local basis of operators to probe the information. We show that when evolving with SU(d)-symmetric random unitaries obeying the Haar distribution,

$$\frac{1}{{2}^{n}}\frac{1}{3}\sum _{{P}_{r}\ne I}{\int}_{{{\mathcal{U}}}_{\times }}dU\,{\rm{Tr}}({\widetilde{W}}^{\dagger }{P}_{r}^{\dagger }\widetilde{W}{P}_{r})=\Omega \left(\frac{1}{{n}^{3/2}}\right).$$

(15)

This contrasts the case with conservation law as first noted by⁴⁶ with random unitaries with energy conservation. One would expect that under symmetry, the operator $\widetilde{W}$ cannot have support over entire systems. For instance, in the case of U(1) symmetry, any conserved quantities would only have non-trivial support on local conserved quantities⁷¹.

We investigate finite-size residual value via OTOC between two 2-local conserved quantities (exchange interactions), which also constitute the smallest non-trivial 2-local SU(d)-symmetric basis elements. Denote ${V}_{r}\equiv (1/\sqrt{{d}^{2}-1}){{\bf{S}}}_{r}\cdot {{\bf{S}}}_{r+1}$ such that

$${\rm{Tr}}({V}_{r}I)=0,{\rm{Tr}}({V}_{r}^{\dagger }{V}_{r})={d}^{n}\,{\rm{and}}\,[{V}_{r},{U}^{\otimes n}]=0.$$

(16)

Then we prove the following lower bound for OTOC at late time:

$$\frac{1}{{d}^{n}}{\int}_{{{\mathcal{U}}}_{\times }}dU\,{\rm{Tr}}({\widetilde{W}}^{\dagger }{V}_{r}^{\dagger }\widetilde{W}{V}_{r})=\Omega \left(\frac{1}{{n}^{{(d+2)}^{2}/2}}\right),$$

(17)

by making explicit use of the expansion of Eq. (2) for SU(d)-symmetric 2-deign as well as well-known facts like S_n characters theory (see SM III.B for more details). In the special case of qubits (d = 2), the lower bound can be further improved to Ω(1/n³).

In both cases, a nonvanishing finite-size residual OTOC is present which only exhibits a power law decay with respect to n. It is interesting to observe that OTOC with respect to the local SU(2)-symmetric basis with local dimension 2 has a faster mode of decay, which may indicate that there is further residual information that is leaked out to the charge sectors which is not probed by the locally symmetric basis. In summary, we lower bound the OTOC of SU(d)-symmetric conserved quantities given by exchanging interactions under both Pauli basis and SU(d)-symmetric local basis. The calculation with respect to the charge density (or equivalently single-quibt Pauli group element), although using a similar strategy such as that under the symmetric basis, is more complicated. It is conceivable that in order to integrate over the group ${{\mathcal{U}}}_{\times }$ of SU(d)-symmetric unitaries, a sensible way is looking at the charge sector decomposition in Eq. (1) and seek to perform the integration on each inequivalent irrep block S^λ. However, since V_r does not obey the SU(d) symmetry, its matrix representation does not fit into the charge sector decomposition and poses a challenge for our computation. We employ mathematical methods used in quantum angular momentum (see SM III.B for more details) for the qubit (d = 2) case and leave the computation of general qudits for future work. Note that these lower bounds are not necessarily tight. Further analysis of the quantum hydrodynamics with non-Abelian conserved quantities would be worthwhile.

Near-optimal SU(d)-covariant codes

Random unitaries exhibit good error-correction and decoupling properties. In the presence of continuous symmetries, U(1) and SU(d)-symmetric random unitaries generate nearly optimal covariant error-correcting codes¹⁵ that they saturate the fundamental limits of the error correction imprecision with scaling $O(\frac{1}{n})$ as identified by the approximate Eastin–Knill theorems (see refs. ^{14,27,28,51,52}). A key feature for non-perfect error correction is due to the fact that there is always logical information leaking into the environment even encoding only one logical qudit and against one-qudit erasure error. We slightly improve the SU(d)-covariant random codes by encoding k logical qudits while still achieving O(1/n) in averaged Choi error asymptotically against single qudit erasure. The actual computation only requires the SU(d)-covariant codes to satisfy a 2-design condition and it is proved in ref. ⁷² that there are symmetric local ensembles capable of converging to this SU(d)-symmetric 2-design in polynomials steps. This provides further motivation to study the SU(d)-covariant random codes, especially on certain natural physical platforms^73,74 since our encoding of Ψ on $\bar{A}$ can be taken as pure states for all local dimension d (see ref. ¹⁸). Concerning quantum information recovery especially in physical contexts, the decoding procedures through e.g. Kitaev–Yoshida decoding protocol in the presence of charge conservation^8,9,10,11 remain to be studied.

We adopt the complementary channel formalism^15,64. We partial trace out B instead of $\bar{B}$ in the discussion around Eq. (7) and obtain

$${\rho }_{R\cup \bar{B}}=\,{{\rm{Tr}}}_{B}(U({\Phi }^{RA}\otimes {\Psi }^{\bar{A}}){U}^{\dagger })$$

(18)

from the encoding protocol. We also assume that ${\Psi }^{\bar{A}}$ is a pure and SU(d)-symmetric state. It is straightforward to check that this defines an SU(d)-covariant encoding map in the following sense:

$$\begin{array}{l}{\hat{U}}^{\otimes n}\left(U({\Phi }^{RA}\otimes {\Psi }^{\bar{A}}){U}^{\dagger }\right){\hat{U}}^{\dagger \otimes n}\\ =\left(U({\hat{U}}^{\otimes k}{\Phi }^{RA}{\hat{U}}^{\dagger \otimes k}\otimes {\Psi }^{\bar{A}}){U}^{\dagger }\right),\end{array}$$

(19)

where ${\hat{U}}^{\otimes k}$ is the transversal action of the group SU(d) acting on A of k qudits.

Then we compute Choi error against the decoupled states $\frac{I}{{d}_{A}}\otimes \zeta$ where ζ is some quantum state in the environment $\bar{B}$ that the erasure error occurs:

$${\epsilon }_{{\rm{Choi}}}=\mathop{\min }\limits_{\zeta }P\left({\rho }_{R\cup \bar{B}},\frac{1}{{d}_{A}}{I}_{R}\otimes \zeta \right)$$

(20)

The quantity $P(\rho ,\sigma )=\sqrt{1-F{(\rho ,\sigma )}^{2}}$ is called purified distance with $F(\rho ,\sigma )=\,{\rm{Tr}}\sqrt{\sqrt{\sigma }\rho \sqrt{\sigma }}$ being the fidelity between two density matrices. The following inequality with the 1-norm distance also holds:

$$\frac{1}{2}\parallel \rho -\sigma {\parallel }_{1}\le P(\rho ,\sigma )\le \sqrt{2\parallel \rho -\sigma {\parallel }_{1}}.$$

(21)

We can bound the Choi error Eq. (20) by triangle inequality:

$$\begin{array}{lll}\;\;\;\mathop{\min }\limits_{\zeta }P\left({\rho }_{R\cup \bar{B}},\frac{1}{{d}_{A}}{I}_{R}\otimes \zeta \right)\\ \le P({\rho }_{R\cup \bar{B}},{\rho }_{R\cup \bar{B},{\rm{avg}}})+\mathop{\min }\limits_{\zeta }P\left({\rho }_{R\cup \bar{B},{\rm{avg}}},\frac{1}{{d}_{A}}{I}_{R}\otimes \zeta \right)\\ \le \sqrt{2\parallel {\rho }_{R\cup \bar{B}}-{\rho }_{R\cup \bar{B},{\rm{avg}}}{\parallel }_{1}}+\mathop{\min }\limits_{\zeta }P\left({\rho }_{R\cup \bar{B},{\rm{avg}}},\frac{1}{{d}_{A}}{I}_{R}\otimes \zeta \right),\end{array}$$

(22)

with ${\rho }_{R\cup \bar{B},{\rm{avg}}}=\int{\rho }_{R\cup \bar{B}}dU$ being averaged over the group of SU(d)-symmetric unitaries. To find the expectation of ϵ_Choi, we integrate the above inequality under SU(d)-symmetric Haar distribution and the integral only uses its first and second moments (1- and 2-designs). Instead of using the partial decoupling theorem⁷⁵, we can analytically compute the first term from the triangle inequality Eq. (22) thanks to the fact that ${\Psi }^{\bar{A}}$ is a pure state, which we show the techniques in SM III.D when k, t = o(n),

$$\begin{array}{lll}\quad{\mathbb{E}}\left[\sqrt{\parallel {\rho }_{R\cup \bar{B}}-{\rho }_{R\cup \bar{B},{\rm{avg}}}{\parallel }_{1}}\right]\le \sqrt{{\mathbb{E}}[\parallel {\rho }_{R\cup \bar{B}}-{\rho }_{R\cup \bar{B},{\rm{avg}}}]{\parallel }_{1}}\\ \le \sqrt{{({d}_{R}{d}_{\bar{B}})}^{1/2}{\left({\mathbb{E}}[{\rm{Tr}}({\rho }_{R\cup \bar{B}}^{2})]-{\rm{Tr}}({\rho }_{R\cup \bar{B},{\rm{avg}}}^{2})\right)}^{1/2}}\\ \le \dfrac{1}{{e}^{\Omega (n)}}.\end{array}$$

(23)

Similar to the familiar Page theorem⁶³, the exponential rate of suppression is achieved, which is due to the selection of relevant charge sectors whose dimension scales exponentially with respect to the system size n (see SM I.B for more details).

If the erasure environment $\bar{B}$ contains a single qudit, we have

$$\begin{array}{lll}{\rho }_{{\rm{avg}}}(R\cup \bar{B})&=&\frac{1}{n}\sum\limits_{a=1}^{k}\frac{1}{{d}^{k-1}}{I}_{R-a}\otimes \left\vert {\Phi }_{a,\bar{B}}\right\rangle \left\langle {\Phi }_{a,\bar{B}}\right\vert \\ &&+\frac{n-k}{n}\frac{1}{{d}^{k}}{I}_{R}\otimes \frac{1}{d}{I}_{\bar{B}},\end{array}$$

(24)

where the state $\left\vert {\Phi }_{a,\bar{B}}\right\rangle$ denotes the maximally entangled Bell pair between the ath qudit in the register R and that erased qudit in $\bar{B}$. As expected, this is a slight generalization of the form given in¹⁵ with k = 1 using a single Bell pair. By choosing $\zeta =\frac{I}{{d}_{\bar{B}}}$ we bound the second term in (22) by several matrix inequalities:

$$\begin{array}{l}{\mathcal{P}}\left({\rho }_{R\cup \bar{B},{\rm{avg}}},\frac{I}{{d}_{A}}\otimes \frac{I}{{d}_{\bar{B}}}\right)\\ \le \sqrt{1-\frac{1}{{d}^{2}}{\left(\sqrt{\frac{k}{n}+\frac{n-k}{n}\frac{1}{{d}^{2}}}+\sqrt{\frac{n-k}{n}\frac{1}{{d}^{2}}}({d}^{2}-1)\right)}^{2}}\\ \approx \frac{k\sqrt{{d}^{2}-1}}{2n}.\end{array}$$

(25)

As expected, the inevitable leakage of information is due to the residual Bell pair coupled between the register R and $\bar{B}$. For erasure beyond one qudit, the entanglement between register R and $\bar{B}$ is likely to take a more complicated form and it would be interesting to investigate the optimality of SU(d)-covariant codes under multiple qudits erasure in future study. For now, we show that with pure state encoding ${\Psi }^{\bar{A}}$, the SU(d)-covariant codes which encode k logical qudits against one qudit erasure are asymptotically optimal covariant codes saturating the scaling limits given by the approximate Eastin–Knill theorems^{14,27,28,51,52}. We note that this upper-bound is tight and it might indicate that for a non-constant coding rate k = O(f(n)), the asymptotic optimality might lose. Indeed, if k = O(f(n)) for a non-constant f(n), we have that

$${\mathcal{P}}({\rho }_{{\rm{avg}}}(R\cup \bar{B})),\frac{I}{{d}^{k}}\otimes \left.\frac{I}{{d}_{\bar{B}}}\right)\ge \alpha \frac{f(n)}{n}$$

(26)

for some constant α. Hence, as shown in Fig. 2, the SU(d) random covariant codes fail to be close to the fundamental limit if the coding rate is non-constant. This stands a sharp contrast to random codes without symmetry, where the error is suppressed exponentially with ${n}_{\bar{A}}-{n}_{A}=n-2k$.

Fig. 2: The log-log plot Purified distance between the decoupled states between R and $\bar{B}$ and ${\rho }_{{\rm{avg}}}(R\cup \bar{B})$ with three logical qubits.

The residual entanglement displayed is a consequence of the Eastin–Knill theorems and imperfect information recovery due to conservation law^10,27,29,30.

Overparametrization regime with geometric quantum machine learning

The third application concerns a general class of geometric quantum machine learning^17,19,54 models where the ansatze respect the underlying symmetry of the problem. With the success of the classical geometric and equivariant machine learning models, there has been a surge of interest in adapting symmetry to quantum machine learning ansatze^{17,18,53,76,77}. These symmetry-respecting or equivariant quantum machine learning ansatze can significantly outperform ones without symmetry in many tasks such as learning ground states of the frustrated antiferromagnetic Heisenberg model^{18,53,78,79,80,81} and weighted graphs⁸². Despite the superior performance and parameter efficiency from empirical observations, it is imperative to ask if there exists a theoretical guarantee for exponential convergence in the number of gradient descent steps or queries. This exponential convergence is highly desirable in the near-term application of QML and variational quantum eigensolver (VQE) in learning complex quantum many-body physics and beyond. For this, we generalize the quantum neural tangent kernel (QNTK) which states that, if the ansatz mimics up to the second moment of the concerned Haar distribution, i.e., achieves the unitary 2-design, then the exponential convergence would arrive at the overparametrization regime.

To begin with, let us consider the variational CQA ansatz U(θ) that respects SU(d) symmetry¹⁸:

$$\begin{array}{l}U(\theta )=\exp (-i{H}_{{\rm{Y\; JM}}}(\beta ))\exp (-i{H}_{S}(\gamma ))\\ =\exp \left(-i\sum\limits_{k,l}{\beta }_{kl}{X}_{k}{X}_{l}\right)\exp \left(-i\gamma \sum\limits_{j=1}^{n-1}(j,j+1)\right),\end{array}$$

(27)

where (j, j + 1) are adjacent SWAPs on qudits and X_k = (1, k) + (2, k) + ⋯ + (k − 1, k) is the so-called Young-Jucys-Murphy element, or YJM-element for short^83,84,85,86, which is essential in the study of SU(d)-symmetric universality theorem as well as k-designs^18,34. The reason to incorporate second-order products X_kX_l of YJM-elements is also explained in detail in these papers. Note that YJM-elements commute with each other, so $\exp (-i{\sum }_{k,l}{\beta }_{kl}{X}_{k}{X}_{l})={\prod }_{k,l}\exp (-i{\beta }_{kl}{X}_{k}{X}_{l})$. Decomposing the second exponential of adjacent SWAPs, however, introduces Trotter errors. A initial state $| {\psi }^{\lambda }\left.\right\rangle =| {\alpha }_{T}^{\lambda },m\left.\right\rangle$ is then taken from one S_n irrep S^λ. We would also consider the statistical ensemble ρ_λ of these states later.

For a given observable $\hat{O}$ that is SU(d)-symmetric, e.g., the Heisenberg Hamiltonian with the form

$$\hat{O}=\sum\limits_{i=1}^{N}{O}_{i},$$

(28)

where each O_i contains a single term. In quantum many-body theory with locality assumption, it is typically the case that the number N of O_i scales linearly or polynomially with the number of qubits. Let the loss be defined as

$$\begin{array}{lll}\quad\;\frac{1}{2}{\left({\rm{Tr}}(U(\theta ){\rho }_{\lambda }{U}^{\dagger }(\theta )\hat{O})-{E}_{0}\right)}^{2}\\ =\;\frac{1}{2}{\left({\rm{Tr}}({\rho }_{\lambda }{U}^{\dagger }(\theta )\hat{O}U(\theta ))-{E}_{0}\right)}^{2}\equiv \frac{1}{2}{\varepsilon }^{2},\end{array}$$

(29)

where E₀ is the ground truth label (normally real-or integer-valued) during a supervised learning process for regression and classification purposes (E₀ might subtly relate to the frozen kernel claim and could trigger phase transitions in quantum machine learning dynamics⁸⁷). QNTK concerns the question of the number of iterations needed in order for a hybrid classical-quantum variational algorithm to converge. In other words, we could define each interaction along with its classical computing resources as a query and minimizing the number of queries is a key aspect in observing any potential quantum advantage. It is shown in refs. ^20,21 that an exponential convergence guarantee can be achieved when

$$\epsilon (t)\approx {(1-\eta K)}^{t}\epsilon (0),$$

(30)

if K does not fluctuate too much around its mean $\bar{K}$ and for learning rate η sufficiently small. The detailed concentration conditions for which the QNTK needs to satisfy are given in refs. ^20,21. In the case where K is sufficiently close to its average case, we say that we have reached the overparametrization regime where an exponential convergence guarantee is observed if the resulting $\eta \bar{K}$ is of order O(1). The threshold of the overparametrization regime accounts for how many variational parameters—a quantity hiding in $\bar{K}$—are needed to achieve this. We show that, very generally, under the assumption that the Hilbert space is decomposed into a direct sum of invariant controlled charge sectors with multiplicities, the overparametrization regime still occurs at O(d_λ) where ${d}_{\lambda }=\dim {S}^{\lambda }$ is the dimension of the charge sector for which the initial states encode.

The following computation actually holds for symmetries governed by general compact groups, e.g., the permutation symmetry defined through permuting qudits by the group S_n or U(1) symmetry, so let us rewrite a generic ansatz as

$$U(\theta )={U}_{-,l}{U}_{+,l},$$

(31)

where l = 1, ⋯ , L is the index for the variational angles. Note that our ansatz can be further Trotterized into products of the time evolution of unitaries and each of which is only parameterized by one variational angle. Then the differentiation with chain rule easily reads:

$$\begin{array}{l}\frac{\partial U(\theta )}{\partial {\theta }_{l}}={U}_{-,l}(-i{H}_{l}){U}_{+,l},\\ \frac{\partial {U}^{\dagger }(\theta )}{\partial {\theta }_{l}}={U}_{+,l}^{\dagger }(i{H}_{l}){U}_{-,l}^{\dagger },\end{array}$$

(32)

where H_l denotes the Hamiltonian generator of the ansatz driven by the parameter θ_l. The QNTK is given by:

$$\begin{array}{lll}K&=&\sum\limits_{l}\frac{\partial \bar{\varepsilon }}{\partial {\theta }_{l}}\frac{\partial \varepsilon }{\partial {\theta }_{l}}\\ &=&\sum\limits_{l}{\left\vert \left\langle \right.{\psi }^{\lambda }| {U}_{+,l}^{\dagger }\left[{H}_{l},{U}_{-,l}^{\dagger }\hat{O}{U}_{-,l}\right]{U}_{+,l}| {\psi }^{\lambda }\left.\right\rangle \right\vert }^{2}\\ &=&-{\left(\left\langle \right.{\psi }^{\lambda }| {U}_{+,l}^{\dagger }\left[{H}_{l},{U}_{-,l}^{\dagger }\hat{O}{U}_{-,l}\right]{U}_{+,l}| {\psi }^{\lambda }\left.\right\rangle \right)}^{2}\\ &=&-{\left(\sum\limits_{i = 1}^{N}\left\langle \right.{\psi }^{\lambda }| {U}_{+,l}^{\dagger }\left[{H}_{l},{U}_{-,l}^{\dagger }{O}_{i}{U}_{-,l}\right]{U}_{+,l}| {\psi }^{\lambda }\left.\right\rangle \right)}^{2},\end{array}$$

(33)

where we assume that the observable $\hat{O}$ respects a certain symmetry whose matrix representation decomposes in a way resembling those in Eq. (1) and $| {\psi }^{\lambda }\left.\right\rangle$ (or ρ_λ) is taken from one charge sector, still labeled by λ for brevity. Due to this choice, we only need to concern the unitary U^λ restricted to that irrep and

$$\begin{array}{lll}K&=&-\sum\limits_{l}\sum\limits_{i,j=1}^{N}{\rm{Tr}}\left({\rho }_{\lambda }{U}_{+,l}^{\lambda \dagger }\left[{H}_{l}^{\lambda },{O}_{i}^{\lambda ,I}\right]\right.\\ &&\left.{U}_{+,l}^{\lambda }{\rho }_{\lambda }{U}_{+,l}^{\lambda \dagger }\left[{H}_{l}^{\lambda },{O}_{j}^{\lambda ,I}\right]{U}_{+,l}^{\lambda }\right),\end{array}$$

(34)

where we define the interaction picture observable ${\hat{O}}^{I}={U}_{-,l}^{\lambda \dagger }\hat{O}{U}_{-,l}^{\lambda }$ (and similarly for ${O}_{i}^{\lambda ,I}$) with general statistical ensemble ρ_λ. The assumption of respecting the symmetry is practical in the experiment which also provides an accessible way to average the above identity by Haar randomness. As explained in the previous context, twirling a general operator M with no assumption on symmetry leads to the most intricate issues when we study OTOC and covariant codes.

We put the detailed computation to obtain the average $\bar{K}$ in SM III.E. In conclusion,

$$\begin{array}{lll}\bar{K}\;=\;-\sum\limits_{l}\int\left(\int\,{\rm{Tr}}\left({\rho }_{\lambda }{U}_{+,l}^{\lambda \dagger }\left[{H}_{l}^{\lambda },{O}^{\lambda ,I}\right]\right.\right.\\\qquad \left.\left.{U}_{+,l}^{\lambda }{\rho }_{\lambda }{U}_{+,l}^{\lambda \dagger }\left[{H}_{l}^{\lambda },{O}^{\lambda ,I}\right]{U}_{+,l}^{\lambda }\right)d{U}_{+,l}\right)d{U}_{-,l}\\\quad\;=\frac{2}{({d}_{\lambda }+1)({d}_{\lambda }^{2}-1)}\left({\rm{Tr}}[{({O}^{\lambda })}^{2}]-\frac{{[{\rm{Tr}}({O}^{\lambda })]}^{2}}{{d}_{\lambda }}\right)\\\qquad\sum\limits_{l}\left({\rm{Tr}}[{({H}_{l}^{\lambda })}^{2}]-\frac{{[{\rm{Tr}}({H}_{l}^{\lambda })]}^{2}}{{d}_{\lambda }}\right).\end{array}$$

(35)

In the special case of SU(d) symmetry, we expand O and H_l by SWAPs and their products in YJM elements. Then their trace restricted to one irrep is just given by S_n group characters^88,89,90. Unfortunately, the character formula becomes intractable for general qudits and arbitrary S_n irreps, but when d = 2 for qubits, we can explicitly check that they scale as Θ(d_λ) for large n. Therefore,

$$\bar{K}\approx \sum\limits_{i,j=1}^{N}\frac{L}{{d}_{\lambda }^{2}}\left({\rm{Tr}}\left({O}_{i}^{\lambda }{O}_{j}^{\lambda }\right)-\frac{{\rm{Tr}}\left({O}_{i}^{\lambda }\right)\,{\rm{Tr}}\left({O}_{j}^{\lambda }\right)}{{d}_{\lambda }}\right)\approx \frac{{N}^{2}L}{{d}_{\lambda }},$$

(36)

where the dimension d_λ of the irrep S^λ can be evaluated by the so-called hook length formula in S_n representation theory^44,55. Some of them scale exponentially with respect to the system size n and we also provide a detailed introduction at the end of SM I.A. Moreover, if one concerns with the permutation symmetry given by permuting qudits through the group S_n, Schur–Weyl duality asserts that the decomposed charge sectors are given by SU(d) irreps whose dimensions scales as O(n^d). Especially when d = 2 for qubits, it is familiar that the largest charge sector, spin-n/2 irrep, preserving permutation invariance is of dimension n + 1.

Disscusion

In this work, we explore the physical applications of random quantum circuits with SU(d) symmetry. We first study the late-time residual of local SU(d) non-Abelian quantities through OTOC, where we show that the residual part decays only inverse polynomials with respect to the total sites n. These asymptotic results stand in agreement with general observation for chaotic dynamics with conservation laws^35,46. The late time (post-scrambling) analysis in our results could motivate a further study of quantum hydrodynamics of quantum circuits with SU(d) conservation laws, where in ref. ³⁴ a local random circuit ensemble is provided for which to converge to unitary k-designs for general qudits. Furthermore, our results may also invite further research in line with studying the entanglement generation through Page curve or short-range chaotic models with non-Abelian quantities of which SU(d)-symmetric quantum circuits could serve as a principal model. It is also worth investigating various SU(d)-symmetric monitored circuit dynamics of local charges such as⁴¹ where novel physical phenomena may appear due to the nature of non-commuting charges. In quantum error correction, as stated in¹⁵, it is of interest to investigate the optimality of random SU(d)-covariant codes against multiple qudit erasure error. Furthermore, it is of interest to find covariant codes that could be capable of a non-constant coding rate. It is also interesting to implement such SU(d)-covariant codes in certain platforms where exchange interaction is naturally implemented^73,74 and logical information is restricted to a certain charge sector. In geometric quantum machine learning our results might give an indication of exponential convergence in the presence of symmetry. For instance, in the presence of permutation symmetry on qubit systems where the charge sectors scale at most O(n), exponential convergence can be quickly achieved by requiring variational parameters linearly in n. It is worth mentioning that many physical interesting problems still bear exponentially scaling charge sectors such as the frustrated Heisenberg models¹⁸. Hence, our work might motivate further study on the performance of these symmetry-respecting variational ansatz in geometric quantum machine learning.