Magic spreading in random quantum circuits

Abstract

Magic is the resource that quantifies the amount of beyond-Clifford operations necessary for universal quantum computing. It bounds the cost of classically simulating quantum systems via stabilizer circuits central to quantum error correction and computation. In this paper, we investigate how fast generic many-body dynamics generate magic resources under the constraints of locality and unitarity, focusing on magic spreading in brick-wall random unitary circuits. We explore scalable magic measures intimately connected to the algebraic structure of the Clifford group. These metrics enable the investigation of the spreading of magic for system sizes of up to N = 1024 qudits, surpassing the previous state-of-the-art, which was restricted to about a dozen qudits. We demonstrate that magic resources equilibrate on timescales logarithmic in the system size, akin to anti-concentration and Hilbert space delocalization phenomena, but qualitatively different from the spreading of entanglement entropy. As random circuits are minimal models for chaotic dynamics, we conjecture that our findings describe the phenomenology of magic resources growth in a broad class of chaotic many-body systems.

Introduction

Quantum computers require several types of resources to solve computational tasks faster than classical computers^1,2. Entanglement is one such resource, but alone, it is insufficient to guarantee that a quantum computer outperforms its classical counterpart. Indeed, stabilizer states can attain extensive entanglement under Clifford operations while being efficiently simulatable on classical computers via the Gottesman-Knill theorem^3,4,5. Nonstabilizerness, colloquially called “magic”, quantifies the additional non-Clifford operations required to perform a given quantum operation, constituting another necessary ingredient for the quantum speedup⁶. Understanding how magic resources build up and propagate in many-body quantum systems emerges as a fundamental question, with potential impact on current and near-term quantum devices⁷.

The question of magic resources generation is ambitious but challenging. Until a few years ago, measures of magic required minimization procedures over large spaces, resulting in prohibitive computational costs for even a few qubits⁸. Recently, mana and stabilizer entropies have been introduced as scalable measures of magic^{9,10,11,12,13}. Subsequent developments in tensor network methods^14,15,16,17 and Monte-Carlo approaches^18,19 have provided a powerful toolbox to characterize the nonstabilizerness of ground states²⁰ while enabling hybrid Clifford-tensor network algorithms^{21,22,23,24,25}. Despite these successes, the time evolution of magic resources in many-body systems remains a largely open question. The rapidly growing entanglement limits the traditional tensor network methods and brute-force exact simulations to small system sizes. With these limitations, Ref. ²⁶ concluded that a quantum quench in an integrable system results in a linear growth of nonstabilizerness over time, similar to what happens for the entanglement entropy²⁷.

This work investigates the magic spreading under generic, non-integrable, local unitary quantum dynamics. To that end, we focus on Haar random brick-wall circuits of qudits. Unambiguous identification of the features of the growth of magic resources requires access to large system sizes. For this reason, inspired by the algebraic structure of the Clifford group, we consider the family of generalized stabilizer entropies (GSE). The latter constitute good measures of nonstabilizerness for many-body systems and include stabilizer Rényi entropy (SRE) as a particular example. We combine the replica trick and Haar average methods to express the circuit-averaged GSE as tensor network contractions. In particular, the GSE can be expressed as a k = 3 replica quantity for qutrits while necessitating k = 4 replica for qubits. In both cases, the replica tensor network methods allow us to investigate systems of up to N ≤ 1024 qubits. Our main finding is that the long-time saturation value of GSE entropy is reached, up to a tolerance ϵ ≪ 1, at times ${t}_{{{{\rm{sat}}}}}^{{{{\rm{mag}}}}}\propto \ln N$, scaling logarithmically with system size, see Fig. 1.

Fig. 1: Cartoon of magic spreading in random circuits.

A system of N qudits is prepared at time t = 0 in a product state $\left\vert {\Psi }_{0}\right\rangle$ with low-magic resources. The evolution under a quantum circuit comprising local Haar-random gates increases the nonstabilizerness of the state (denoted by the red gradient) and scrambles quantum information (symbolized by the light-cone). The nonstabilizerness approaches its long-time saturation value up to a given tolerance ϵ ≪ 1 at time ${t}_{{{{\rm{sat}}}}}^{{{{\rm{mag}}}}}\propto \ln N$, scaling logarithmically with N, while distant qudits become entangled only after a longer time ${t}_{{{{\rm{sat}}}}}^{{{{\rm{ent}}}}}\propto N$.

Results

Generalized stabilizer entropies

We start by discussing the GSE, characterizing the magic resources for systems of N qudits^10,28. Denoting ${{\mathbb{Z}}}_{d}=\{0,1,\ldots,d-1\}$ the finite field with d elements, the local Hilbert space of a d-dimensional qudit is ${{{{\mathcal{H}}}}}_{d}={{{\rm{span}}}}[{\{\left\vert m\right\rangle \}}_{m\in {{\mathbb{Z}}}_{d}}]$. The GSEs are tied to the algebraic structure of the Pauli and Clifford groups³ acting on ${{{{\mathcal{H}}}}}_{d}^{\otimes N}$. Let us denote the qudit Pauli operators

$$X={\sum}_{m=0}^{d-1}\left\vert m\right\rangle \left\langle m{\oplus }_{d}1\right\vert,\qquad Z={\sum}_{m=0}^{d-1}{\omega }^{m}\left\vert m\right\rangle \left\langle m\right\vert,$$

(1)

with $a{\oplus }_{d}b=a+b\,{{{\rm{mod}}}}\,d$ representing the sum in ${{\mathbb{Z}}}_{d}$ and ω = e^2πi/d 29. Then, the set of Pauli strings ${{{{\mathcal{P}}}}}_{N}(d)=\{{X}_{1}^{{r}_{1}^{x}}{Z}_{1}^{{r}_{1}^{z}}{X}_{2}^{{r}_{2}^{x}}{Z}_{2}^{{r}_{2}^{z}}\cdots {X}_{N}^{{r}_{N}^{x}}{Z}_{N}^{{r}_{N}^{z}}\,| \,{r}_{k}^{\alpha }\in {{\mathbb{Z}}}_{d}\}$ is generated by tensor products of Pauli operators. The Clifford group ${{{{\mathcal{C}}}}}_{N,d}$ consists of the unitary C mapping, up to a global phase, a Pauli string P to a Pauli string ${\omega }^{r}{P}^{{\prime} }=CP{C}^{{{\dagger}} }$ with $r\in {{\mathbb{Z}}}_{d}$. Stabilizer states are defined as ${{{{\rm{STAB}}}}}_{N,d}=\{C{\left\vert 0\right\rangle }^{\otimes N}\,| \,C\in {{{{\mathcal{C}}}}}_{N,d}\}$, and magic or non-stabilizer states are those not belonging to this set.

The theory of GSE is intimately connected to the structure of the commutants of the Clifford group for k copies of the system^30,31, cf. Supplementary Note 1 for a brief review. For a given set ${{{\mathcal{E}}}}$, we define its k-th commutant as the set of operators W acting on k copies of the system such that ${{{{\rm{Comm}}}}}_{k}({{{\mathcal{E}}}})=\{W\,| \,[W,{E}^{\otimes k}]=0\,\,{\mbox{for all}}\,E\in {{{\mathcal{E}}}}\}$. For the unitary group ${{{\mathcal{U}}}}({d}^{N})$, the commutant is built of the representation of permutation operators ${{{{\rm{Comm}}}}}_{k}({{{\mathcal{U}}}}({d}^{N}))=\{{W}_{\pi }\,| \,\pi \in {S}_{k}\}$³². By duality, since ${{{{\mathcal{C}}}}}_{N,d}\subset {{{\mathcal{U}}}}({d}^{N})$, it follows that ${{{{\rm{Comm}}}}}_{k}({{{\mathcal{U}}}}({d}^{N}))\subset {{{{\rm{Comm}}}}}_{k}({{{{\mathcal{C}}}}}_{N,d})$. Importantly, the Clifford commutant contains $| {{{{\rm{Comm}}}}}_{k}({{{{\mathcal{C}}}}}_{N,d})|={\prod }_{m=0}^{k-2}({d}^{m}+1)$ elements, whereas $| {{{{\rm{Comm}}}}}_{k}({{{\mathcal{U}}}}({d}^{N}))|=k!$. These two numbers coincide for d > 2 when k ≤ 2, and for d = 2 when k ≤ 3. Hence, the elements of the commutant, when applied to a state ${\left(\left\vert \Psi \right\rangle \left\langle \Psi \right\vert \right)}^{\otimes k}$ cannot distinguish whether $\left\vert \Psi \right\rangle$ is a stabilizer state or whether it has non-vanishing magic resources if k ≤ 3 for qubits and k ≤ 2 for d ≥ 3.

On the other hand, when k ≥ 3 for qudits (and k ≥ 4 for qubits), the Clifford commutant is strictly greater than ${{{{\rm{Comm}}}}}_{k}({{{\mathcal{U}}}}({d}^{N}))$. In that case, the intrinsic Clifford commutant²⁸, defined as ${\overline{{{{\rm{Comm}}}}}}_{k}({{{{\mathcal{C}}}}}_{N,d})= {{{{\rm{Comm}}}}}_{k}({{{{\mathcal{C}}}}}_{N,d})\setminus {{{{\rm{Comm}}}}}_{k}({{{\mathcal{U}}}}({d}^{N}))$, is a non-empty set, whose elements W, when applied to ${\left(\left\vert \Psi \right\rangle \left\langle \Psi \right\vert \right)}^{\otimes k}$ can distinguish if the state is a stabilizer state or not. We define the generalized stabilizer entropy, M_W, and the associated generalized stabilizer purity ζ_W by

$${M}_{W}\equiv -\ln [{\zeta }_{W}(\left\vert \Psi \right\rangle )],\quad {\zeta }_{W}\equiv {{{\rm{tr}}}}(W\left\vert \Psi \right\rangle {\left\langle \Psi \right\vert }^{\otimes k})\,.$$

(2)

For any $W\in {\overline{{{{\rm{Comm}}}}}}_{k}({{{{\mathcal{C}}}}}_{N,d})$, the generalized stabilizer purity ${\zeta }_{W}(\left\vert \Psi \right\rangle )\le 1$, with the equality holding if and only if $\left\vert \Psi \right\rangle$ is a stabilizer state, as we show in the Supplementary Note 2. This implies that the GSE ${M}_{W}(\left\vert \Psi \right\rangle )\ge 0$, with the equality holding if and only if $\left\vert \Psi \right\rangle$ is a stabilizer state. Moreover, for any operator W from the intrinsic Clifford commutant, the associated GSE M_W is a measure of magic for many-body systems: (i) ${M}_{W}(\left\vert \Psi \right\rangle )\ge 0$ and M_W = 0 if and only if $\left\vert \Psi \right\rangle$ is a stabilizer state, (ii) ${M}_{W}(C\left\vert \Psi \right\rangle )={M}_{W}(\left\vert \Psi \right\rangle )$ for $C\in {{{{\mathcal{C}}}}}_{N,d}$ a Clifford unitary, (iii) it is additive ${M}_{W}(\left\vert \Phi \right\rangle \otimes \left\vert \Psi \right\rangle )= {M}_{W}(\left\vert \Phi \right\rangle )+{M}_{W}(\left\vert \Psi \right\rangle )$. The subclass of stabilizer Rényi entropies are monotones for generic stabilizer protocols for qubits¹². A key contribution of our work is showing that stabilizer entropies are monotone also for prime dimension qudits (d ≥ 3), as we discussed in Methods. In the Supplementary Note 3 we also argue that monotonicity under Pauli measurements of arbitrary GSE is not guaranteed, presenting an explicit counterexample for qutrits systems.

To understand better the scope of GSE, we first consider how SRE arise from (2). At k = 4, the intrinsic Clifford commutant contains several operators, one of which is ${Q}_{4}={\sum }_{P\in {{{{\mathcal{P}}}}}_{N}(d)}{(P\otimes {P}^{{{\dagger}} })}^{\otimes 2}/{d}^{N}$. This operator leads to the second SRE¹⁰${M}_{2}\equiv {M}_{{Q}_{4}}=-\ln [{\zeta }_{{Q}_{4}}]$ with ${\zeta }_{{Q}_{4}}={\sum }_{P\in {{{{\mathcal{P}}}}}_{N}(d)}| \left\langle \Psi \right\vert P\left\vert \Psi \right\rangle {| }^{4}/{d}^{N}$. Similarly, the SRE M_α of arbitrary integer index α ≥ 2, ${M}_{\alpha }\equiv \ln [{\sum }_{P}| \left\langle \Psi \right\vert P\left\vert \Psi \right\rangle {| }^{2\alpha }/{d}^{N}]/(1-\alpha )$, fulfills ${M}_{\alpha }\propto {M}_{{Q}_{2\alpha }}$ with ${Q}_{2\alpha }={\sum }_{P\in {{{{\mathcal{P}}}}}_{N}(d)}{(P\otimes {P}^{{{\dagger}} })}^{\otimes \alpha }/{d}^{N}$. With a slight abuse of notation, throughout this text, we will consider ${M}_{2}\equiv {M}_{{Q}_{4}}$. An example of GSE beyond the family of SREs is by the operator ${Y}_{d}\equiv {\sum }_{P\in {{{{\mathcal{P}}}}}_{N}(d)}(P\otimes P\otimes {P}^{d-2})/{d}^{N}$, which belongs to the intrinsic Clifford commutant for k = 3 replicas in any qudit system with d ≥ 3 prime. Throughout this paper, we denote by M_Y the GSE induced by the operator Y_d, with the dimension d inferred from context.

The above examples provide concrete measures of the magic M_W that require k = 3 copies for qudits with odd prime d and k = 4 for qubits. A figure of merit valid for any stabilizer purity ζ_W is their efficient tensor network representation. In fact, the Clifford commutant operators T acting on N qudit systems, reduce to tensor products of operators W = w^⊗N acting on individual qudits. As a result, Eq. (2) is efficiently computable via tensor network methods, either by exact contractions^14,17 or by sampling methods¹⁶, see Supplementary Note 4 for details.

Brick-wall Haar random quantum circuits

We consider a one-dimensional chain of N qudits and study the spreading of the GSEs M_W under unitary dynamics generated by brick-wall Haar random quantum circuits (see Fig. 1). The evolution operator of the considered brick-wall circuit reads ${U}_{t}={\prod }_{r=1}^{t}{U}^{(r)}$, where t is the circuit depth – also referred to as time. Numbering the qudits by i = 1, . . . , N, the layers U^(r) are fixed as

$${U}^{(2m)}=\mathop{\prod }_{i=1}^{N/2-1}{U}_{2i,2i+1},\quad {U}^{(2m+1)}=\mathop{\prod}_{i=1}^{N/2}{U}_{2i-1,2i},$$

(3)

comprising two-qubit gates U_i,j chosen independently with the Haar distribution on the unitary group ${{{\mathcal{U}}}}({d}^{2})$. The initial state $\left\vert {\Psi }_{0}\right\rangle$ is chosen as the stabilizer state $\left\vert {\Psi }_{0}\right\rangle={\left\vert 0\right\rangle }^{\otimes N}$, with M_W = 0 for any W in the intrinsic Clifford commutant. How do the magic resources of $\left\vert {\Psi }_{t}\right\rangle$, quantified by the GSEs, increase under the dynamics of the circuit (3)?

The problem at hand is stochastic due to the randomness of the gates. Denoting with ${\mathbb{E}}(\bullet )$ the average over the circuit realizations, we consider quenched and annealed averages of the GSEs, defined respectively as $\overline{{M}_{W}}\equiv {\mathbb{E}}[{M}_{W}(\left\vert {\Psi }_{t}\right\rangle )]=-{\mathbb{E}}[\ln [{\zeta }_{W}(\left\vert {\Psi }_{t}\right\rangle )]]$ and ${\tilde{M}}_{W}\equiv -\ln [{\mathbb{E}}[{\zeta }_{W}(\left\vert {\Psi }_{t}\right\rangle )]]$. Exact numerical simulation of the random circuit (3) provides access to $\left\vert {\Psi }_{t}\right\rangle$, allowing for calculation of the quenched and the annealed averages of the GSEs and exposing the self-averaging of ${\zeta }_{W}(\left\vert {\Psi }_{t}\right\rangle )$. The circuit-to-circuit fluctuations of ${\zeta }_{W}(\left\vert {\Psi }_{t}\right\rangle )$ around its average value ${\mathbb{E}}[{\zeta }_{W}(\left\vert {\Psi }_{t}\right\rangle )]$ are suppressed with the increase of the system size N and decay rapidly in time as discussed in Methods for several choices of W. The self-averaging of ${\zeta }_{W}(\left\vert {\Psi }_{t}\right\rangle )$ implies that $\overline{{M}_{W}}$ and ${\tilde{M}}_{W}$ approach each other with increase of N and t. Therefore, the annealed average ${\tilde{M}}_{W}$ may be chosen to quantify the time evolution of the GSEs under the random circuits.

Annealed average of generalized stabilizer entropies

The calculation of the annealed average ${\tilde{M}}_{W}$ is facilitated by a replica trick and the Weingarten calculus, which allow us to map computation of ${\tilde{M}}_{W}$ for the random Haar circuits to a contraction of a two-dimensional tensor network. The latter can be efficiently computed to provide insights into the time evolution of the GSEs for systems comprising hundreds of qudits, far beyond the reach of exact simulation of the system.

For convenience, we employ the superoperator formalism³³: $A\mapsto \left.\left\vert A\right\rangle \right\rangle$, ${U}_{t}A{U}^{{{\dagger}} }\mapsto ({U}_{t}\otimes {U}_{t}^{*})\left.\left\vert A\right\rangle \right\rangle$, and $\left\langle \left\langle \right.\right.A\left.\left\vert B\right\rangle \right\rangle={{{\rm{tr}}}}({A}^{{{\dagger}} }B)$. Using the common abuse of notation of implicit reshaping ${({{{{\mathcal{H}}}}}_{d}^{\otimes N})}^{\otimes k}\mapsto {({{{{\mathcal{H}}}}}_{d}^{\otimes k})}^{\otimes N}$ when necessary, we have

$${\zeta }_{W}(\left\vert {\Psi }_{t}\right\rangle )=\left\langle \left\langle \right.W\right\vert {({U}_{t}\otimes {U}_{t}^{*})}^{\otimes k}\left.\left\vert {\rho }_{0}^{\otimes k}\right\rangle \right\rangle,$$

(4)

where ${\rho }_{0}=\left\vert {\Psi }_{0}\right\rangle \left\langle {\Psi }_{0}\right\vert$ is the initial state’s density matrix. To calculate the average ${\mathbb{E}}[{\zeta }_{W}(\left\vert {\Psi }_{t}\right\rangle )]$ over the circuit realizations, we observe that (4) is linear in ${({U}_{t}\otimes {U}_{t}^{*})}^{\otimes k}$, implying that we can first average the superoperator corresponding to the circuit and then calculate the matrix element in (4). Due to the statistical independence of the two-body gates at various spatial and temporal locations, the former reduces to evaluating the two qudit transfer matrix ${{{{\mathcal{T}}}}}_{i,i+1}^{(k)}\equiv {{\mathbb{E}}}_{{{{\rm{Haar}}}}}[{({U}_{i,i+1}\otimes {U}_{i,i+1}^{*})}^{\otimes k}]$. These expressions are formulated in terms of the unitary commutant ${{{{\rm{Comm}}}}}_{k}({{{\mathcal{U}}}}({d}^{N}))$ which, as aforementinoed, consists of permutation operators. Up to reshaping, we express ${{{{\mathcal{T}}}}}^{(k)}$ via the corresponding permutation states ${\left.\left\vert \tau \right\rangle \right\rangle }_{i}$ acting on the k-replica qudits at sites i, i + 1, leading to the expression

$${{{{\mathcal{T}}}}}_{i,i+1}^{(k)}={\sum}_{\pi,\tau \in {S}_{k}}{{{{\rm{Wg}}}}}_{\tau,\sigma }({d}^{2}){\left.\left\vert \tau \right\rangle \right\rangle }_{i}{\left.\left\vert \tau \right\rangle \right\rangle }_{i+1}{\left\langle \left\langle \right.\sigma \right\vert }_{i}{\left\langle \left\langle \right.\sigma \right\vert }_{i+1},$$

(5)

where $\left\langle \left\langle \right.\right.{b}_{1},{\bar{b}}_{1},\ldots,{b}_{k},{\bar{b}}_{k}\left.\left\vert \tau \right\rangle \right\rangle={\prod }_{m=1}^{k}{\delta }_{{b}_{m},{\bar{b}}_{\tau (m)}}$ for each k-replica qudit basis state $\left.\left\vert {b}_{1},{\bar{b}}_{1},\ldots,{b}_{k},{\bar{b}}_{k}\right\rangle \right\rangle$ and Wg_τ,σ(d²) denotes the Weingarten symbol³⁴. The lattice structure induced by the circuit requires contraction of ${{{{\mathcal{T}}}}}_{i,i+1}^{(k)}$ between the even and odd layers (3), with the overlaps ${G}_{\sigma,\tau }(d)\equiv \left\langle \left\langle \right.\right.\sigma \left.\left\vert \tau \right\rangle \right\rangle={d}^{\#({\sigma }^{-1}\tau )}$ taken into account, where #(τ) denotes the number of cycles for τ ∈ S_k. We reabsorb these overlaps by defining the tensors

(6)

with the states $\left.\left\vert \hat{\sigma }\right\rangle \right\rangle$ satisfying $\left\langle \left\langle \right.\right.\hat{\sigma }\left.\left\vert \tau \right\rangle \right\rangle={\delta }_{\sigma,\tau }$. The contraction with the first layer of unitary gates is fixed by the replica boundary condition

(7)

while the contraction with the last layer of the circuit requires

(8)

Summarizing, the computation of the annealed average of the GSEs reduces to evaluating the tensor contraction

(9)

The effective “spins”, i.e., the degrees of freedom at the sites of the lattice (9), correspond to permutations of the k replicas and hence admit q_eff = k! values, while the tensors ${{{{\mathcal{T}}}}}_{i,i+1}^{(k)}$ can be interpreted as non-unitary gates acting on the spins. These observations constitute the basis of our numerical approach, which allows us to compute the annealed average of the GSEs ${\tilde{M}}_{W}=-\ln [{\mathbb{E}}[{\zeta }_{W}(\left\vert {\Psi }_{t}\right\rangle )]]$ for arbitrary circuit depth t. While the above discussion applies to any intrinsic Clifford commutant operator W, we will specifically analyze the SRE M₂ for k = 4 replicas in qubit (d = 2) and qutrit (d = 3) systems, and M_Y for k = 3 in qutrit systems.

Deep circuit limit

In the deep circuit limit, for t ≫ 1, the brick-wall quantum circuits form approximate k-designs^35,36. In that limit, the operator U_t in (4) can be replaced by a global Haar random gate $U\in {{{\mathcal{U}}}}({d}^{N})$ and ${{{{\mathcal{T}}}}}_{i,i+1}^{(k)}$ in the contraction (9) are substituted by the global gate. This allows for analytical calculation of the ${\tilde{M}}_{W}$ of interest as detailed in the Supplementary Note 5, yielding ${M}_{2}^{{{{\rm{Haar}}}}}\equiv -\ln [4/({2}^{N}+3)]$ for d = 2³⁷, while for d = 3, we find ${M}_{2}^{{{{\rm{Haar}}}}}={M}_{Y}^{{{{\rm{Haar}}}}}\equiv -\ln [3/({3}^{N}+2)]$. Due to the concentration of Haar measure³², the fluctuations of M_W with circuit realizations are strongly suppressed with the increase of N. Hence, the GSEs ${\tilde{M}}_{W}$ saturate at long times t ≫ 1 under the dynamics of random circuits to ${M}_{W}^{{{{\rm{Haar}}}}}$. Now, we characterize the approach of ${M}_{W}(\left\vert {\Psi }_{t}\right\rangle )$ to the saturation value ${M}_{W}^{{{{\rm{Haar}}}}}$.

Numerical results

Our results for the growth of GSEs under the dynamics of random circuits are summarized in Figs. 2–4 for qubits (d = 2) and qutrits (d = 3), respectively. We start by comparing the quenched ${\tilde{M}}_{W}$ and annealed ${\overline{M}}_{W}$ averages of the GSEs. As anticipated, already for N = 8 qubits and qudits, we find ${\tilde{M}}_{W}\approx {\overline{M}}_{W}$, confirming that the quenched and annealed averages can be used interchangeably to characterize magic spreading in the considered circuits, cf. Methods for details. Hence, we focus on the annealed averages ${\tilde{M}}_{W}$ obtained from the tensor network contraction (9). Expressing the state of the q_eff = k! dimensional “spins” as a matrix product state³⁸, we contract the tensor network (9) horizontally, layer after layer. Implementing the contraction in ITensor³⁹, we observe that a bond dimension $\chi={{{\mathcal{O}}}}({q}_{{{{\rm{eff}}}}}^{2})$ of the matrix product state is sufficient to obtain converged results, see Methods. The computation requires significantly smaller resources for qutrits since M_Y is a k = 3-replica quantity resulting in q_eff = 6. In contrast, the k = 4 replicas demanded to calculate M₂ for qubits lead to q_eff = 24, and significantly larger computational costs. In this case, to alleviate computational complexity, we use irreducible representations of the permutation group, effectively reducing the replica degrees of freedom to q_eff = 14⁴⁰. We compute ${\tilde{M}}_{2}$ for systems of up to N = 1024 qubits (N = 512 qutrits) with χ = 300 (χ = 800), as shown in Fig. 2 (Fig. 3), and ${\tilde{M}}_{Y}$ for N ≤ 1024 qutrit systems with χ = 300, cf. Fig. 4.

Fig. 2: Stabilizer Rényi entropy evolution in qubit circuits.

a The SRE M₂ abruptly saturates to ${M}_{2}^{{{{\rm{Haar}}}}}$. b The difference $\Delta {M}_{2}={M}_{2}^{{{{\rm{Haar}}}}}-{M}_{2}$ approaches exponential decay $\Delta {M}_{2}\propto N{e}^{-{\alpha }_{2,2}t}$, where α_2,2 = 0.43(3), see the inset. The annealed average ${\tilde{M}}_{2}$ obtained via (9) (denoted “TN'') and the quenched average ${\overline{M}}_{2}$ (denoted “ED'') coincide within the error bars already for N = 8.

Fig. 3: Stabilizer Rényi entropy evolution in qutrit circuits.

a The saturation of M₂ to ${M}_{2}^{{{{\rm{Haar}}}}}$ for d = 3 occurs similarly to the qubit case. b The difference $\Delta {M}_{2}={M}_{2}^{{{{\rm{Haar}}}}}-{M}_{2}$ follows $\Delta {M}_{2}\propto N{e}^{-{\alpha }_{3,2}t}$ with α_3,2 = 1.03(3) at t ≳ 5; see the inset. The quenched ${\tilde{M}}_{2}$ and annealed ${\overline{M}}_{2}$ averages are indistinguishable on the scale of the figure for any N.

Fig. 4: Dynamics of the generalized stabilizer entropy M_Y for qutrits circuits.

a The saturation of M_Y to ${M}_{Y}^{{{{\rm{Haar}}}}}$ occurs similarly to the qubit case. b The difference $\Delta {M}_{Y}={M}_{Y}^{{{{\rm{Haar}}}}}-{M}_{Y}$ follows $\Delta {M}_{Y}\propto N{e}^{-{\alpha }_{3,Y}t}$ with α_3,Y = 0.98(2) at t ≳ 5; see the inset. The quenched ${\tilde{M}}_{Y}$ and annealed ${\overline{M}}_{Y}$ averages approach each other with the increase of N.

In all the cases, the GSEs ${\tilde{M}}_{W}$ are proportional to the system size N already at t = 1. Indeed, the additivity of ${\tilde{M}}_{W}$ implies that ${\tilde{M}}_{W}(\left\vert {\Psi }_{t=1}\right\rangle )=(N/2){\tilde{M}}_{W}^{(2)}$, where ${\tilde{M}}_{W}^{(2)}$ is the average GSE generated by a single two-body gate U_i,i+1, and N/2 is the number of two-body gates in the first layer of the circuit. For t > 1, the GSEs ${\tilde{M}}_{W}$ rapidly increase towards their saturation values ${M}_{W}^{{{{\rm{Haar}}}}}$ for both d = 2 and d = 3. For circuit depths t ≳ 5, the difference $\Delta {M}_{W}(t)={M}_{W}^{{{{\rm{Haar}}}}}-{\tilde{M}}_{W}(\left\vert {\Psi }_{t}\right\rangle )$ is proportional to the system size N and decays exponentially in time:

$$\Delta {M}_{d}(t)={a}_{d,W}N{e}^{-{\alpha }_{d,W}t},$$

(10)

where a_d,W and α_d,W are constants (see Figs. 2–4). The exponential relaxation of the GSEs to their long-time saturation values under the dynamics of random quantum circuits is the main result of this work. The saturation value of GSEs is reached, up to a fixed small accuracy ϵ, i.e., ΔM_W = ϵ, at time ${t}_{{{{\rm{sat}}}}}^{{{{\rm{mag}}}}}=\ln (N)/{\alpha }_{d,W}+O(1)$, scaling logarithmically with system size N.

Discussion

The brick-wall Haar random quantum circuits combine principles of locality and unitarity of time dynamics, serving as minimal models for ergodic quantum many-body systems⁴¹. Random circuits provide insights into the entanglement growth^27,42,43, the properties of operator spreading^44,45,46 and spectral correlations^47,48,49. In particular, the magic resources in eigenstates of ergodic many-body systems share properties with states of deep random circuits³⁷. Hence, we conjecture that the universal features of the GSEs growth (10), i.e., the exponential relaxation to the saturation value at times ${t}_{{{{\rm{sat}}}}}^{{{{\rm{mag}}}}}\propto \ln N$ characterize the spreading of GSEs in chaotic many-body systems. Our conjecture is supported by the following observation based on the Suzuki-Trotter decomposition⁵⁰. Consider quantum dynamics generated by a local ergodic quantum Hamiltonian H = ∑_jH_j,j+1. The Suzuki-Trotter formula is a key element of algorithms computing time dynamics of many-body system, e.g., the time-evolving block decimation⁵¹, and allows for the following approximation of the evolution operator

$${e}^{-i\Delta tH}\approx {\prod }_{k=1}^{N/2}{e}^{-i\Delta t{H}_{2k-1,2k}}\mathop{\prod}_{k=1}^{N/2-1}{e}^{-i\Delta t{H}_{2k,2k+1}},$$

(11)

valid for sufficiently small Δt. Eq. (11) reproduces the structure of the brick-wall quantum circuit, with unitary gates given by ${U}_{k,k+1}^{H}={e}^{-i\Delta t{H}_{k,k+1}}$. The gates ${U}_{k,k+1}^{H}$ for generic ergodic many-body systems do not belong to the Clifford group, and their action increases the nonstabilizerness of the state of the system. Hence, each layer of the circuit defined by (11) contains extensively many gates that increase magic resources, similar to the brick-wall Haar random quantum circuits considered in this work.

The uncovered phenomenology of the GSEs parallels, as we argue in Supplemental Note 6, the growth of mana⁸ under the dynamics of random quantum circuits, even though mana is a magic state resource theory monotone not belonging to the family of GSEs. The behavior of these nonstabilizerness measures is reminiscent of time evolution of participation entropy⁵², which characterizes the spread of many-body states in a selected basis of the Hilbert space and saturates at times ${t}_{{{{\rm{sat}}}}}^{({{{\rm{pe}}}})}\propto \ln N$⁵³. The latter is tied to anticoncentration of the state of logarithmically deep random circuits^54,55, which is a necessary assumption of the formal proofs underlying quantum advantage. The rapid growth of nonstabilizerness is in a stark contrast with the ballistic increase of entanglement entropy under ergodic many-body dynamics^27,56, resulting in a saturation timescale ${t}_{{{{\rm{sat}}}}}^{({{{\rm{ent}}}})}\propto N$, linear in system size. At a formal level, the difference arises due to the disparity in boundary conditions at the top layer of the circuit corresponding to the GSEs (9) and the entanglement entropy⁴² calculations. Physically, the time required to entangle two distant regions by local quantum dynamics scales linearly with the separation between the regions, consistent with the scaling of ${t}_{{{{\rm{sat}}}}}^{{{{\rm{ent}}}}}$. In contrast, the GSEs capture global properties of the state, and already time ${t}_{{{{\rm{sat}}}}}^{{{{\rm{ent}}}}}\propto \ln N$ is sufficient for nonstabilizerness to equilibrate even though entanglement between the most distant qudits has not yet been generated.

In summary, in this paper we have explored the dynamics of magic resources focusing on brick-wall Haar random unitary circuits. To that end, we considered the GSEs M_W as scalable measures of nonstabilizerness, which include, but are not limited to, the SRE. Our investigations reveal that magic resources are rapidly generated by the dynamics of random unitary circuits and saturate at relatively short times which scale logarithmically with the system size. The revealed behavior of M_W aligns with the log-depth anticoncentration of random quantum circuits⁵⁴ and matches the phenomenology of Hilbert space delocalization under random circuits⁵³. The GSE spreading remains qualitatively different from the ballistic growth of entanglement entropy in ergodic many-body systems. Since the random circuits constitute a minimal model of local unitary dynamics, we expect a similar phenomenology of magic state resources evolution to arise in generic ergodic many-body systems.

Understanding how the phenomenology of nonstabilizerness generation changes when the ergodicity is broken due to, e.g., many-body localization^57,58 or quantum scars⁵⁹, is an open question. Steps in that direction were already taken for integrable systems^26,60,61, and doped Clifford circuits⁶² in which the generation of the magic resources is slower due to sparseness of beyond-Clifford operations, cf. the Supplementary Note 6. The asymmetry between the generation of magic resources and entanglement by local dynamics provides a new perspective onto the relation of entanglement and magic phase transitions^63,64,65,66. The framework based on the algebraic structure of the Clifford group, which yielded the GSEs, hosts more examples of magic measures with potential for better characterization of magic in many qudit systems. We leave these problems open for further research.

Methods

Numerical simulations

We employ two complementary numerical approaches: (i) exact circuit simulation and (ii) tensor network contraction for the annealed average of GSE. Together, they provide crucial insights into magic spreading in random quantum circuits. Exact simulation reveals the self-averaging properties of the SRE and GSE, but is limited to small system sizes. Exploiting this fact, we use tensor network contraction to compute annealed averages, which accurately approximate quenched averages, as we argue in the following.

Self-averaging

Exact numerical simulation of quantum circuits’ dynamics involves generating the computational basis ${{{\mathcal{B}}}}=\{\left\vert {{{{\bf{x}}}}}_{1}{{{{\bf{x}}}}}_{2}\cdots {{{{\bf{x}}}}}_{N}\right\rangle \,| \,{{{{\bf{x}}}}}_{j}\in {{\mathbb{Z}}}_{d}\}$ and expressing the state $\left\vert {\Psi }_{t}\right\rangle$ as a superposition of the computational basis states. Then, the action of two-body Haar-random gates on states in ${{{{\mathcal{H}}}}}_{N,d}$ reduces to sparse matrix-vector multiplication. The first limiting factor of the exact simulation is the exponential growth of $| {{{\mathcal{B}}}}|={d}^{N}$ with the system size. A more severe constraint arises from the need to evaluate $| {{{{\mathcal{P}}}}}_{N}(d)|={d}^{2N}$ Pauli string expectation values in the time-evolved state $\left\vert {\Psi }_{t}\right\rangle$ to compute the GSEs of interest, cf. Main Text. We developed and employed an efficient numerical algorithm, allowing these exact calculations to system sizes up to N = 22 qubits and N = 12 qutrits.

We employ the exact numerical simulation to calculate the quenched average of the GSEs ${\overline{M}}_{W}=-{\mathbb{E}}[\ln [{\zeta }_{W}(\left\vert {\Psi }_{t}\right\rangle )]]$ and the annealed average ${\tilde{M}}_{W}=-\ln [{\mathbb{E}}[{\zeta }_{W}(\left\vert {\Psi }_{t}\right\rangle )]]$, where the circuit average involves 1000 realizations unless otherwise specified. Additional details are presented in the Supplementary Note 6.

Focusing on the difference $\delta {M}_{W}(t)=| {\overline{M}}_{W}(t)-{\tilde{M}}_{W}(t)|$ between the quenched and annealed averages of SRE and GSE finding that

$$\delta {M}_{2}(t)={a}_{t,2}N+{b}_{t},$$

(12)

where a_t,2 and b_t,2 are constant at fixed time t. This behavior characterizes δM₂(t) both for qubits (d = 2) and qutrits (d = 3). We note that analogous behavior holds for M_Y(t). After an initial transient at small circuit depths t, the coefficients a_t,W decrease exponentially over time for all relevant operators W in both qubit and qutrit systems

$${a}_{t,W}={a}_{W}{e}^{-{\beta }_{d,W}t},$$

(13)

where β_d,W is a constant dependent on the on-site Hilbert space dimension d and on W. For qubits, we find β_2,2 = 0.83(3) while for qutrits β_3,2 = β_3,Y = 1.97(5). These values of β_d,W confirm that the error made when the quenched averages are interchanged with the annealed averages is negligible for sufficiently large N and t. Indeed, the exponential decay of ΔM_W(t), as described in Eq. (10), occurs at a rate α_2,2 = 0.43(3) for qubits and α_3,2 = 1.05(3) for qutrits for M₂, while, for qutrits, ΔM_Y(t) decays with a rate of α_3,Y = 0.98(2). The rates β_d,W are significantly (approximately twice) larger than the corresponding rates α_d,W. Hence, the relative errors committed when the quenched and annealed averages are interchanged decays exponentially in time as

$$\delta {M}_{W}(t)/\Delta {M}_{W}(t)\propto {e}^{-({\beta }_{d,W}-{\alpha }_{d,W})t},$$

(14)

up to a sub-leading in system size term O(1/N). For any t ≥ 0, the relative error is smaller than 3% in all the considered cases and a clear exponential decay of the relative error is observed at t ≿ 5–10. All the details are included in the Supplementary Note 6.

The above scaling analysis demonstrates that approximating quenched averages with the annealed averages in the dynamics of brick-wall quantum circuits is justified at any time t and that this approximation improves exponentially with t as shown by (14). Moreover, our numerical results indicate that the linear scaling (12) is robust and persists at any system size N. These two trends show that our main conclusion about the magic resources growth in random quantum circuits, i.e. the logarithm depth saturation of magic resources, ${t}_{{{{\rm{sat}}}}}^{{{{\rm{mag}}}}}\propto \ln (N)$, is accurate in the scaling limit of large system size N.

Tensor network contractions

Our tensor network approach aims at an efficient evaluation of the tensor network contraction (9) which yields the annealed average ${\tilde{M}}_{W}(t)$. To that end, we interpret (9) as a non-unitary time evolution of a state of N effective “spins” with on-site Hilbert space dimension q_eff = k! growing factorially with the number k of replicas. The non-unitary time evolution is followed by the contraction of the obtained state with the last layer (8).

The state of the effective spins is expressed as a matrix product state (MPS)

$$\left.\left\vert \Pi \right\rangle \right\rangle={\sum}_{{\tau }_{1},\ldots,{\tau }_{N}\in {S}_{k}}{A}_{[1]}^{{\tau }_{1}}{A}_{[2]}^{{\tau }_{2}}\cdots {A}_{[N]}^{{\tau }_{N}}\left.\left\vert {\tau }_{1},\ldots,{\tau }_{N}\right\rangle \right\rangle,$$

(15)

where $\left.\left\vert {\tau }_{1},\ldots,{\tau }_{N}\right\rangle \right\rangle=\left.\left\vert {\tau }_{1}\right\rangle \right\rangle \otimes \left.\left\vert {\tau }_{2}\right\rangle \right\rangle \cdots \otimes \left.\left\vert {\tau }_{N}\right\rangle \right\rangle$ is the product state of representations of permutations τ_i ∈ S_k while ${A}_{[i]}^{{\tau }_{i}}$ are χ × χ matrices for any i = 2, …, N–1, ${A}_{[1]}^{{\tau }_{1}}$ (${A}_{[N]}^{{\tau }_{N}}$) are 1 × χ (χ × 1) matrices and χ is the bond dimension, which needs to be contrasted with the on-site local Hilbert dimension q_eff. At t = 1, the state of the system $\left.\left\vert {\Pi }_{1}\right\rangle \right\rangle$ is the product state (7).

The time evolution consists of contraction of the state of the system $\left.\left\vert {\Pi }_{t}\right\rangle \right\rangle$ with subsequent layers of the tensors ${{{{\mathcal{T}}}}}_{i,i+1}^{(k)}$ defined by (6). The contraction of ${{{{\mathcal{T}}}}}_{i,i+1}^{(k)}$ with the matrices ${A}_{[i],{a}_{i},{a}_{i+1}}^{{\tau }_{i}}$, ${A}_{[i+1],{a}_{i+1},{a}_{i+2}}^{{\tau }_{i+1}}$ (where a_i, a_i+1, a_i+2 denote the matrix indices of A_[i],  A_[i+1]) results in a tensor ${{{{\mathcal{T}}}}}_{{a}_{i},{a}_{i+2}}^{{\tau }_{i},{\tau }_{i+1}}$. Reshaping the tensor to ${{{{\mathcal{T}}}}}_{({\tau }_{i},{a}_{i}),({\tau }_{i+1},{a}_{i+2})}$ results in a matrix of dimension (q_eff χ) × (q_eff χ), which is expressed back as a product of two matrices ${A}_{[i]}^{{\prime} },\,{A}_{[i+1]}^{{\prime} }$ via the standard singular value decomposition (SVD). Notably, the dimension of the matrix ${{{{\mathcal{T}}}}}_{({\tau }_{i},{a}_{i}),({\tau }_{i+1},{a}_{i+2})}$ is increased by a factor q_eff with respect to the bond dimension χ. The complexity of SVD scales as ${({q}_{{{{\rm{eff}}}}}\chi )}^{3}$ which is the main problem hindering our calculations for qubits (d = 2), since the on-site Hilbert space dimension is q_eff = 4! = 24. This fact was one of our motivations for introducing the GSE M_Y and considering qutrits, which results in a much smaller on-site Hilbert space dimension q_eff = 3! = 6 and a significantly simpler computation of the tensor network contraction. Finally, for qubits (d = 2), as noted in ref. ⁴⁰, the effective on-site Hilbert space dimension can be reduced from q_eff = k! to ${q}_{{{{\rm{eff}}}}}^{{\prime} }={C}_{k}$, where C_k is the Catalan number. This enables us to calculate ${\tilde{M}}_{2}(t)$ for qubits by considering model with ${q}_{{{{\rm{eff}}}}}^{{\prime} }={C}_{4}=14$ dimensional on-site Hilbert space. To find the mapping between the tensors of (9) living q_eff = k! dimensional space to the ${q}_{{{{\rm{eff}}}}}^{{\prime} }=14$ subspace, we employed the single qubit Haar-random gate which, upon averaging, corresponds to the projector onto the 14-dimensional invariant subspace.

The results presented in the Main Text are converged with the bond dimension χ of the MPS (15). The convergence occurs once $\chi \approx {q}_{{{{\rm{eff}}}}}^{2}$, see Supplementary Note 6 for additional details.

Results beyond generalized stabilizer entropies: mana

In the Main Text, we considered the growth of GSEs under dynamics of Haar random brick-wall circuits acting on systems of qubits and qutrits finding the exponential relaxation of these nonstabilizerness measures towards their long time values (10). Here, we argue that analogous phenomenology is shared by mana, a nonstabilizerness monotone defined for odd prime d as the negativity of the Wigner representation of the state ρ⁸.

The phase-space point operators A_r are defined in terms of the Pauli strings ${P}_{{{{\bf{r}}}}}={\prod }_{l=1}^{N}{X}_{l}^{{r}_{l}^{x}}{Z}_{l}^{{r}_{l}^{z}}$ (where ${{{\bf{r}}}}\equiv ({r}_{1}^{x},{r}_{1}^{z},\ldots,{r}_{N}^{x},{r}_{N}^{z})\in {{\mathbb{Z}}}_{d}^{2N}$) as

$${A}_{{{{\bf{0}}}}}=\frac{1}{{d}^{N}}{\sum}_{{{{\bf{r}}}}\in {{\mathbb{Z}}}_{{{{\bf{d}}}}}^{2N}}{P}_{{{{\bf{r}}}}},\quad {A}_{{{{\bf{r}}}}}={P}_{{{{\bf{r}}}}}{A}_{{{{\bf{0}}}}}{P}_{{{{\bf{r}}}}}^{{{\dagger}} }.$$

(16)

The phase-space point operators are orthogonal, ${{{\rm{tr}}}}({A}_{{{{\bf{r}}}}}{A}_{{{{{\bf{r}}}}}^{{\prime}}})={d}^{N}{\delta }_{{{{\bf{r}}}},{{{{\bf{r}}}}}^{{\prime} }}$, enabling to represent the state as

$$\rho=\mathop{\sum}_{{{{\bf{r}}}}\in {{\mathbb{Z}}}_{{{{\bf{d}}}}}^{2N}}\frac{1}{{d}^{N}}{{{\rm{tr}}}}({A}_{{{{\bf{r}}}}}\rho ){A}_{{{{\bf{r}}}}}.$$

(17)

The coefficients of the expansion define the discrete Wigner function ${W}_{\rho }({{{\bf{r}}}})=\frac{1}{{d}^{N}}{{{\rm{tr}}}}({A}_{{{{\bf{r}}}}}\rho )$⁹. Mana ${{{\mathcal{M}}}}$ is defined as the negativity of the Wigner function

$${{{\mathcal{M}}}}=\ln ({\sum}_{{{{\bf{u}}}}}| {W}_{\rho }({{{\bf{u}}}})| ).$$

(18)

Mana offers insights into magic state resources as it is a strong nonstabilizerness monotone both for pure and mixed states.

Performing exact numerical simulations of Haar-random brick-wall circuits for qutrits and calculating mana, we observe that ${{{\mathcal{M}}}}\propto N$ already at t = 1, after a single layer of the circuit. Additionally, ${{{\mathcal{M}}}}$ quickly saturates with t to the value ${{{{\mathcal{M}}}}}^{{{{\rm{Haar}}}}}$ of the Haar-random state of N qutrits. The difference $\Delta {{{\mathcal{M}}}}={{{{\mathcal{M}}}}}^{{{{\rm{Haar}}}}}-{{{\mathcal{M}}}}(t)$ follows an exponential decay with circuit depth t, $\Delta {{{\mathcal{M}}}}={b}_{n}{e}^{-\alpha t}$, with the prefactor b_n increases linearly with N and α converging to a constant with increase of N, see the Supplementary Note 6 for further details. These numerical results suggest that $\Delta {{{\mathcal{M}}}}\propto N{e}^{-\alpha t}$, analogously to the GSEs (10). This demonstrates the universality of the uncovered phenomenology of magic spreading among different measures of nonstabilizerness.

Stabilizer Rényi entropies are magic monotone for qudits

A key contribution of this paper is proving that stabilizer Rényi entropies are magic monotone under general stabilizer protocols for any qudit with dimension d prime. Following ref. ¹², we require only to bound the expectation value ${P}_{\alpha }(\Psi )\equiv {d}^{-N}{\sum }_{P\in {{{{\mathcal{P}}}}}_{N}}| \left\langle \Psi \right\vert P\left\vert \Psi \right\rangle {| }^{2\alpha }$ for a state on N qudits $\left\vert \Psi \right\rangle={\sum }_{i=0}^{d-1}\sqrt{{p}_{i}}\left\vert i\right\rangle \otimes \left\vert {\phi }_{i}\right\rangle$ with ${\sum }_{i=0}^{d-1}{p}_{i}=1$, ${\{\left\vert i\right\rangle \}}_{i=0,\ldots,d-1}$ on-site state in the computational basis, and ${\{\left\vert {\phi }_{i}\right\rangle \}}_{i=0,\ldots,d-1}$ generic states on the remaining N − 1 qudits. In particular, we must show the following Lemma holds:

Theorem 1

Consider $\left\vert \Psi \right\rangle=\mathop{\sum }_{i=0}^{d-1}\sqrt{{p}_{i}}\left\vert i\right\rangle \otimes \left\vert {\phi }_{i}\right\rangle$ and $\left\vert {\phi }_{i}\right\rangle \in {{\mathbb{C}}}^{d\otimes (N-1)}$. For any integer α ≥ 2 it holds that

$${P}_{\alpha }(\Psi )\le \mathop{\max }_{j}\{{P}_{\alpha }({\phi }_{j})\}\,.$$

(19)

We sketch the proof of the lemma, closely following the technique in Ref. ¹², and present the full details in the Supplementary Note 3 A. The key technique is to expand P_α(Ψ) over $\tilde{P}\otimes P$ with $P\in {{{{\mathcal{P}}}}}_{N-1}$ and $\tilde{P}\in {{{{\mathcal{P}}}}}_{1}$

$${P}_{\alpha }(\Psi )=\mathop{\sum}_{\begin{array}{c}P\in {{{{\mathcal{P}}}}}_{N-1}\\ \tilde{P}\in {{{{\mathcal{P}}}}}_{1}\end{array}}\frac{1}{{d}^{N}}{\left| {\sum}_{i,j=0}^{d-1}\left\langle i\right\vert \tilde{P}\left\vert j\right\rangle \left\langle {\phi }_{i}\right\vert P\left\vert {\phi }_{j}\right\rangle \sqrt{{p}_{i}{p}_{j}}\right| }^{2\alpha },$$

(20)

and to explicitly the sum over $\tilde{P}$ using one qudit Pauli operator. After straightforward but lengthly algebra based on the binomial expansion and repeated applications of Cauchy-Schwarz and Hölder inequalities, we obtain

$${P}_{\alpha }(\Psi )\le {\max }_{m}\left\{{P}_{\alpha }({\phi }_{m})\right\}{\sum}_{| {{{\bf{i}}}}|,| {{{\bf{j}}}}|=\alpha }\left(\begin{array}{c}\alpha \\ {{{\bf{i}}}}\end{array}\right)\,\left(\begin{array}{c}\alpha \\ {{{\bf{j}}}}\end{array}\right){\tilde{\delta }}_{{{{\bf{i}}}},{{{\bf{j}}}}}{F}_{{{{\bf{i}}}},{{{\bf{j}}}}}({{{\bf{p}}}}),$$

(21)

where $| {{{\bf{i}}}}|={\sum }_{m=0}^{d-1}{i}_{m}$, ${F}_{{{{\bf{i}}}},{{{\bf{j}}}}}({{{\bf{p}}}})\equiv \mathop{\sum }_{a=0}^{d-1}\mathop{\prod }_{m=0}^{d-1}{({p}_{j\oplus a}{p}_{j})}^{{j}_{m}+{i}_{m}}$, $\left(\begin{array}{c}\alpha \\ {{{\bf{i}}}}\end{array}\right)=\alpha !/({i}_{0}!{i}_{1}!\ldots {i}_{d-1}!)$ is the multinomial coefficient, and ${\tilde{\delta }}_{{{{\bf{i}}}},{{{\bf{j}}}}}\equiv {\delta }_{{\sum }_{m}m({j}_{m}-{i}_{m})=0{{{\rm{mod}}}}d}$ is enforced by the phases of the $\tilde{P}$ operators. Nevertheless, since F_i,j(p) ≤ d^1−2α for all i and j, using standard properties of the multinomial coefficient, it follows that ${P}_{\alpha }(\Psi )\le \mathop{\max }_{m}\{{P}_{\alpha }({\phi }_{m})\}$ as required.