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(1)On the distribution of the mean energy in the unitary orbit of quantum states

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(1)On the distribution of the mean energy in the unitary orbit of quantum states. arXiv:2012.14342v3 [quant-ph] 28 Jul 2021. Raffaele Salvia1 and Vittorio Giovannetti2 1. Scuola Normale Superiore and University of Pisa, I-56127 Pisa, Italy. 2. NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56127 Pisa, Italy. Given a closed quantum system, the states that can be reached with a cyclic process are those with the same spectrum as the initial state. Here we prove that, under a very general assumption on the Hamiltonian, the distribution of the mean extractable work is very close to a gaussian with respect to the Haar measure. We derive bounds for both the moments of the distribution of the mean energy of the state and for its characteristic function, showing that the discrepancy with the normal distribution is increasingly suppressed for large dimensions of the system Hilbert space.. 1 Introduction Haar-uniform random unitary matrices are a resource required for various quantum algorithm [1–3]. As an example, the randomised benchmark protocol is a method to test the error rate of a quantum circuit, requiring it to perform a sequence of random operations [4]. Versions of the randomised benchmark are used by the companies IBM [5] and Microsoft [6] to test the functionality of their experimental quantum computing hardware. Other applications of random unitaries include quantum cryptography [7] and the simulation of many body physics [8, 9]. In this paper we charachterise the the distribution of the energetic cost of implementing a random unitary transformation. To be more specific, we study the distribution of the expected value of the energy gained from the cycle which sends the quantum state ρ to U ρU † , where U is a matrix extracted randomly with respect to the Haar measure of the unitrary group. Our main finding is that, when the quantum system has many degrees of freedom (that is when the Hilbert space has dimension d  1), then -provided a weak condition on the system Hamiltonian- the distribution of the energy cost of a random unitary matrix is approximately Gaussian. Implementing a random unitary is not an easy task. Actually, only a small subset of quantum operations (called the Clifford gates [10]) are easy to implement in an actual circuit - the complexity of a quantum circuit is often measured with the number of nonClifford gates it requires [11]. To overcome these difficulties, it has been theorised the possibility of circuits that simulate a random unitary up to a certain moment of the distribution [12]. A circuit which is able to emulate a uniform distribution up to the t-th moment is called an unitary t-design. If we want to realise a t-design with t > 3, it is still necessary to use non-Clifford gates [13, 14], but only a small amount of them [15]. Raffaele Salvia: Corresponding author: raffaele.salvia@sns.it. Accepted in. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 1.

(2) The results of this paper are useful for characterising an ideal source of uniformly distributed unitary operations. Furthermore, they are valid also for its approximations which are used in actual quantum computing, the t-designs which match the uniform unitary distribution up to the t-th moment. Indeed we will show that, if the dimension of the Hilbert space is big enough, and if it holds a very general condition, the first moments of the distribution are very well approximated by the moments of a gaussian distribution with the same variance. Exploiting this result, we will then be able to estimate the maximum error that we make in replacing the charachteristic function of distribution of the energy with the characteristic function of a normal distribution. We emphasize that our paper is concerned with the distribution of the mean estracted work between different processes, which is different from the distribution of extracted work in a given process (a much more studied subject in in classical and quantum theormodynamics [16, 17]). The latter distribution is in many cases Gaussian for classical systems in the quasistatic limit [18–20], but not quantum system [21]: in particular, as shown in Ref. [22], for slowly driven quantum-thermodynamical processes the work distribution becomes non-gaussian whenever quantum coherences are generated during the protocol. However, we may think to a “process” which consists in selecting a random unitary matrix according to the Haar measure and then applying it to the system: in this case, under the conditions mentioned above the work distribution would be Gaussian: we think it worth to notice that, when the initial state of the system is a state of thermal equilibrium (so that Jarzynski’s equality [23] holds), this implies the validity of the classical fluctuation-dissipation relation [16].. 2 Introduction to the problem Let A be a d-dimensional quantum system initialized in the state ρ̂ and forced to evolve in time by external modulations of its Hamiltonian Ĥ. Following [24, 25], the average amount of work one can extract from the process can be computed as WU (ρ̂; Ĥ) := E(ρ̂; Ĥ) − E(Û ρ̂Û † ; Ĥ) ,. (1). where E(ρ̂; Ĥ) := Tr[ρ̂Ĥ] ,. (2). is the mean energy of ρ̂, and Û is the element of the unitary group U(d) associated to the applied driving. The allowed values of WU (ρ̂; Ĥ) are limited by the inequalities A(ρ̂; Ĥ) ≤ WU (ρ̂; Ĥ) ≤ E(ρ̂; Ĥ) ,. (3). where E(ρ̂; Ĥ) and A(ρ̂; Ĥ) are respectively the ergotropy and anti-ergotropy functionals of the model. The first one was introduced in Ref. [26] and corresponds to maximum work one can extract from the system by optimizing WU (ρ̂; Ĥ) with respect to the choices of the control unitary Û for fixed ρ̂ and Ĥ. A closed formula for the ergotropy is provided by the expression E(ρ̂; Ĥ) := max WU (ρ̂; Ĥ) = WÛ (↓) (ρ̂; Ĥ) U. =. E(ρ̂; Ĥ) − E(ρ̂(↓) ; Ĥ) ,. (4). obtained by setting Û equal to the optimal unitary Û (↓) for which the final state of the cycle Û ρ̂Û † corresponds to the passive counterpart ρ̂(↓) of ρ̂ [24, 25], i.e. d X (↑). (5). Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 2. |j ihλj | ,. j=1. Accepted in. (↓). d X (↓) (↑). (↑). Û (↓) :=. ρ̂(↓) :=. λj |j ihj | ,. j=1.

(3) (↓). (↓). with |λj i the eigenvector of ρ̂ associated with the eigenvalue λj order (i.e.. (↓) λj. ≥. (↓) λj+1 ),. and. in non-decreasing order (i.e.. listed in non-increasing. (↑) (↑) |j i the eigenvector of Ĥ with eigenvalue j listed instead (↑) (↑) j ≤ j+1 ). Similarly for the anti-ergotropy we can write. A(ρ̂; Ĥ) := min WU (ρ̂; Ĥ) = WÛ (↑) (ρ̂; Ĥ) U. =. E(ρ̂; Ĥ) − E(ρ̂(↑) ; Ĥ) ,. (6). where now ρ̂(↑) is the anti-passive state of ρ̂, obtained by the unitary Û (↑) that reverses the order in which the populations of ρ̂(↓) are listed, i.e. Û (↑) :=. d X (↑). (↑). |j ihλj | ,. ρ̂(↑) :=. j=1 (↑). d X (↑) (↑). (↑). λj |j ihj | ,. (7). j=1. (↓). with λj := λd−j+1 . Saturating the upper bound (3) is an important optimization task which can be practically difficult to implement, as it implicitly requires an exact knowledge of the full spectral decomposition of the system Hamiltonian. In this perspective, it is interesting to understand how close one can get from the boundary values (3) by randomly selecting Û for a model in which both ρ̂ and Ĥ are assigned. Clearly, as the dimensionality of the system increases, we do not expect such a naive approach to be particularly effective. Still, providing an exact characterization of the associated efficiency is a well-posed statistical question which can be of some help in identifying which physical systems are best suited as successful candidates for implementing quantum battery models [27–31]. In order to tackle this issue, in the present paper we study the probability distribution P (E|ρ̂; Ĥ) of the mean output energy E := E(Û ρ̂Û † ; Ĥ) ,. (8). which originates by random sampling Û on the unitary group U(d) via its natural measure (the Haar measure dµ(Û )) [32]. A numerical example of this distribution, with d = 7 is showed in Fig. 1. From Eq. (1) it is clear that knowing P (E|ρ̂; Ĥ) we can then reconstruct the probability distribution Pwork (W |ρ̂; Ĥ) of the average extracted work W := WU (ρ̂; Ĥ) via a simple shift of the argument, i.e. . . Pwork (W |ρ̂; Ĥ) := P E = E(ρ̂; Ĥ) − W ρ̂; Ĥ .. (9). Our main finding is that, under mild assumptions on the system Hamiltonian, when the dimension of the Hilbert space d is sufficiently large, the central moments of the probability (µ,Σ(2) ) P (E|ρ̂; Ĥ) can be approximated by those of a gaussian distribution PG (E) having (2) mean value µ and variance Σ equal to those of P (E|ρ̂; Ĥ), i.e. (p). Σ(p) := h(E − µ)p i ' ΣG ,. (10). where hf (E)i denotes the mean value of the function f (E) with respect to P (E|ρ̂; Ĥ) [32] and (p) ΣG. Accepted in. := Gp ×.    1. (p even) ,.   0. (p odd) ,. (11). Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 3.

(4) Figure 1: Numerical estimation of the distribution P (E|ρ̂; Ĥ) in a seven-levels (d = 7) system in which Ĥ has eigenvalues {−1.6, −1.2, −0.6, 0, 0.4, 1.3, 1.7}, and ρ̂ has eigenvalues {0.395, 0.224, 0.151, 0.115, 0.079, 0.0020, 0.013}. The histogram plots the empirical distribution of E(Û ρ̂Û † ; Ĥ) for a sample of 105 unitary matrices U ∈ U(7), which are distributed uniformly according to the Haar measure. The blue line is the probability density function of the Gaussian distribution (µ,Σ(2) ) PG (E) with the variance Σ(2) given by (32). In this example Σ(2) ' 0.02024 and ηĤ ' 0.2317.. Accepted in. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 4.

(5) with Gp the scaling factor . Gp := (p − 1)!! Σ(2). p/2. .. (12). In order to settle the approximation (10) into firm quantitative ground we prove that for d sufficiently larger than p the discrepancies between the l.h.s and the r.h.s of such equation can be bounded as (p). Σ(p) − ΣG. ≤ Gp fĤ (d, p) ,. (13). where fĤ (d, p) is a positive function which in the large d limit scales as O(1/d + η∆Ĥ ) for √ p even, and O( η∆Ĥ ) for p odd, with η∆Ĥ being a functional (see Eq. (74) below) that for typical choices of the system Hamiltonian is very much depressed. Using the theory of generating functions, we also show a stronger result which, aside from bounding the error on each individual moment of the distribution P (E|ρ̂; Ĥ), directly links its characteristic (µ,Σ(2) ). function with the one of the gaussian function PG (E). The rest of the manuscript is organised as follows. Section 3 introduces the concepts and some known results we will use in the proof. In particular, Sec. 3.2 sets up the problem of calculating the moments of the distribution, which can be expressed as the Haar integral (20). The evaluation of this integral can be set up using the theory of Weingarten calculus, that we introduce in section 3.4. The practical computation of the integrals involved in the proof will require other combinatorical concepts, which will be introduced at the points where they are needed. In Sec. 4 we derive some useful inequalities and discuss the assumptions on the system that are needed to enforce the Gaussian approximation. Section 5 presents a proof of Eq. (13) that applies for the special case in which the input state of the system ρ̂ is pure, and in Sec. 6 we address instead the case of arbitrary input states. Finally, in Sec. 7 we bound instead the distance between the characteristic function he−itE i of the distribution P (E|ρ̂; Ĥ), and the characteristic (µ,Σ(2) ). function of the Gaussian distribution PG given in Sec. 8.. (E). Conclusion and outlook are finally. 3 Preliminary considerations This section is dedicated to clarify some useful mathematical properties of the model.. 3.1 Basic properties of P (E|ρ̂; Ĥ) From Eqs. (2) and (3) it follows that for fixed ρ̂ and Ĥ the range of the random variable E is limited by the inequalities Tr[ρ̂(↓) Ĥ] =. d X (↓) (↑). λj j. j=1. ≤E≤. d X (↑) (↑). λj j. = Tr[ρ̂(↑) Ĥ] .. (14). j=1. In the special cases where either the input state is completely mixed (i.e. ρ̂ = 1̂/d), or the Hamiltonian is proportional to the identity (i.e. Ĥ = E0 1̂), the upper and lower bounds of (14) coincide forcing E to assume the constant value Tr[Ĥ]/d, i.e. imposing the distribution P (E|ρ̂; Ĥ) to become a Dirac delta P (E|1̂/d; Ĥ) = P (E|ρ̂; E0 1̂) = δ(E − Trd[Ĥ] ) . Accepted in. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. (15) 5.

(6) From Eq. (8) it is also clear that for arbitrary choices of ρ̂ any rigid shift of the Hamiltonian spectrum results in a translation of the distribution P (E|ρ̂; Ĥ), i.e. P (E|ρ̂; Ĥ) = P E − E0 ρ̂; Ĥ − E0 1̂ . . . (16). More generally noticing that for all choices of ∆0 , E0 ∈ R one has E(Û ρ̂Û † ; Ĥ) = Tr Û ρ̂ − ∆0 1̂d Û † Ĥ − E0 1̂ h. . . +∆0. . . i. . Tr[Ĥ] − E + E , 0 0 d. (17). we can draw the following formal identity P (E|ρ̂; Ĥ). (18). 1̂ E − ∆0 Trd[Ĥ] − E0 − E0 ρ̂ − ∆0 ; Ĥ − E0 1̂ d . =P. !. . ,. where, generalizing the definition of P (E|ρ̂; Ĥ), given  and B̂ generic operators, we use the symbol P (E 0 |Â; B̂) to represent the distribution of the variable E 0 := Tr[Û ÂÛ † B̂] induced by the Haar measure dµ(Û ). We observe next that any two input states ρ̂0 and ρ̂ which have the same spectrum will have the same energy probability distribution, i.e. P (E|ρ̂0 ; Ĥ) = P (E|ρ̂; Ĥ). Indeed under this condition we can always express ρ̂0 as V̂ ρ̂V̂ † with V̂ ∈ U(d), so that for all functions f (E) one has Z. dEf (E)P (E|ρ̂0 ; Ĥ) = Z. 0. 0. Z. dµ(Û )f (E(U V̂ ρ̂V̂ † U † ; Ĥ)). 0†. Z. dµ(Û )f (E(Û ρ̂Û ; Ĥ)) =. =. dEf (E)P (E|ρ̂; Ĥ) ,. where we used the fact that Û 0 = Û V̂ ∈ U(d), and the invariance property dµ(Û ) = dµ(Û V̂ ) of the Haar measure. Similarly due to the cyclicity of the trace appearing in Eq. (2), we can conclude that P (E|ρ̂; Ĥ) is also invariant under unitary rotations of the system Hamiltonian, leading to the identity P (E|V ρV † ; W HW † ) = P (E|ρ̂; Ĥ) ,. ∀V, W ∈ U(d) ,. (19) (↓). which ultimately implies that P (E|ρ̂; Ĥ) can only depend upon the spectra {λj }i and (↑). (↓). {j }i of ρ̂ and Ĥ but not on the specific choices of their associated eigenstates {|λj i}i (↑). and {|j i}i nor on the relative overlap between them.. 3.2 Central moments In evaluating the moments of the distribution P (E; ρ̂, Ĥ) we have to consider the quantities p. hE i =. Z. X. =. j1 ,···jp k1 ,··· ,kp. × Accepted in. Z. h. ip. ρj1 k1 · · · ρjp kp. X. . dµ(Û ) Tr Û ρ̂Û † Ĥ. (20) H`1 i1 · · · H`p ip. `1 ,···`p i1 ,··· ,ip. dµ(Û ) Ui1 j1 Ui2 j2 · · · Uip jp Uk†1 `1 Uk†2 `2 · · · Uk†p `p ,. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 6.

(7) where ρjk , H`i , Uij are matrix elements of ρ̂, Ĥ, Û with respect to fixed basis of the Hilbert space of the system [33]. The integral appearing on the last term of (20) is widely known and admit solution [34–36] in terms of the Weingarten functions C[σ] for the unitary group [34, 37], i.e. Z. dµ(Û ) Ui1 j1 Ui2 j2 · · · Uip jp Uk†1 `1 Uk†2 `2 · · · Uk†p `p X. =. C[σ]. p Y. δia `τ (a) δja kτ σ(a) ,. (21). a=1. τ,σ∈Sp. with Sp representing the permutation group of p elements. A precise definition of the C[σ] s and their properties will be given in Sec. 3.4. Here we simply notice that replacing (21) into (20) leads to the expression X. hE p i =. C[σ] ρ[στ ] H[τ ] ,. (22). τ,σ∈Sp. where for Θ̂ generic operator and σ ∈ Sp , we introduced the functional Θ[σ] :=. X. Θk1 kσ(1) Θk2 kσ(2) · · · Θkp kσ(p) .. (23). k1 ,··· ,kp. To get a more intuitive understanding of what is this product, we need to introduce an important property of a permutation σ ∈ Sp : the structure of its cycles, which is sufficient to specify its conjugacy class [σ]. If the permutation σ has a fixed point, i.e. if there exists an i ∈ {1, 2, · · · , p} such that σ(i) = i, then we say that the permutation σ has a cycle of length 1. A cycle of length 2 (i.e. a transposition) means that there exist two indices i, j ∈ {1, 2, · · · , p} such that σ(i) = j and σ(j) = i. In general given σ ∈ Sp its conjugacy class [σ] is uniquely identified via the correspondence (1). (2). (c[σ]). [σ] ←→ {α[σ] , α[σ] , . . . , α[σ] } ,. (24). with c[σ] being the number of independent cycles admitted by σ, and with the positive (j) integers α[σ] ’s representing instead the lengths of such cycles organized in decreasing order, (j). (j+1). i.e. α[σ] ≥ α[σ] . Because each element in {1, 2, · · · , p} belongs to one and only one of the cycles of σ, we must have that c[σ] X (j). α[σ] = p ,. (25). j=1 (j). implying that the α[σ] s provide a proper partition of the integer p (represented graphically with a Young diagram [38]). The set of indices {1, 2, · · · , p} can thus be partitioned in c[σ] (j) subsets c(i) , which each c(i) having α[σ] elements. We finally observe that two permutations σ and σ 0 will belong to the same conjugacy class if and only if we can identify a third permutation that allows us to relate them via conjugation, i.e. [σ] = [σ 0 ]. Accepted in. ⇐⇒. ∃τ ∈ Sp |. σ 0 = τ στ −1 .. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. (26). 7.

(8) The fact that each element i ∈ {1, 2, · · · , p} belongs to exactly one cycle of σ allows us to rewrite Eq. (23) as c[σ]. X. Θ[σ] =. Y Y. Θki kσ(i). (27). k1 ,··· ,kp r=1 i∈c(r) c[σ]. c[σ]. =. Y. Y. X. r=1 {ki }. i∈c(r). Y. Θki kσ(i) =. . (1). α[σ]. . ,. Tr Θ̂. r=1. i∈c(r). where in the last identity we used the fact that the the r-th summation runs over a set (r) of α[σ] cyclical indices. Equation (27) makes it clear that, as explicitly indicated by the notation, the terms Θ[σ] (as well as the coefficients C[σ] , see Eq. (44) below) depend upon σ only via its conjugacy class [σ]. Replaced into Eq. (22), the identity (27) also implies that hE p i can be expressed as linear combination of products of traces of powers of ρ̂ and Ĥ, i.e. explicitly  X. hE p i =. c[στ ]. C[σ] . Y. (i) α[στ ]. Tr[ρ̂. . c[τ ]. ] . Y. i=1. τ,σ∈Sp. Tr[Ĥ. (j) α[τ ]. . ] .. (28). j=1. Equation (28) is the starting point of our analysis: we notice incidentally that it confirms by virtue of Specht’s theorem [39], that P (E|ρ̂; Ĥ) (and hence its moments) depends upon ρ̂ and Ĥ only through their spectra. In particular for the cases p = 1, 2 we get hEi = C[1] Tr[Ĥ] ,. (29) h. i. hE 2 i = C[1,1] (Tr[Ĥ])2 + C[2] (Tr ρ̂2 )(Tr[Ĥ])2 h. (30). i. +C[1,1] (Tr[Ĥ 2 ])(Tr ρ̂2 ) + C[2] (Tr[Ĥ 2 ]) , which using the explicit values of the functions C[σ] reported in Tab. 1 leads to µ := hEi = Tr[Ĥ]/d , Σ. (2). (31). 2. := h(E − µ) i =. d2. (32). 1 Tr[Ĥ 2 ] − −1. (Tr[Ĥ])2. !. d. h. i. Tr ρ̂2 −. 1 . d . 3.3 Shifting the spectrum of Ĥ Equation (29) makes it clear that, irrespectively from the specific form of the input state of the system ρ̂, we can enforce the distribution P (E; ρ̂, Ĥ) to have zero mean value by setting Tr[Ĥ] = 0 via a rigid shift of the associated spectrum. In particular setting E0 = µ in Eq. (16) we can write . P (E|ρ̂; Ĥ) = P E − µ ρ̂; ∆Ĥ. . ,. (33). while, setting E0 = µ and ∆0 = 1 in Eq. (18) we get . P (E|ρ̂; Ĥ) = P E − µ ∆ρ̂; ∆Ĥ. . ,. (34). ∆Ĥ := Ĥ − µ1̂ ,. (35). where ∆ρ̂ := ρ̂ − Accepted in. 1̂ d. ,. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 8.

(9) are zero-trace operators. Following the derivation of the previous section we can then use Eqs. (33) and (34) to write the central moments of P (E|ρ̂; Ĥ) in the following equivalent forms Σp = h(E − µ)p i =. X. C[σ] ρ[στ ] ∆H[τ ]. (36). C[στ −1 ] ∆ρ[σ] ∆H[τ ] ,. (37). τ,σ∈Sp. =. X τ,σ∈Sp. where ∆ρ[σ] and ∆H[τ ] are now the functional (27) associated with the operators (35), and where in Eq. (37) we changed the summation variable via the introduction of the inverse τ −1 of the permutation τ . Both Eqs. (36) and (37) offer us a huge simplification in the analysis of the problem, with the first having an important application in the special case of pure input states. In particular we notice that since ∆Ĥ is a traceless operators, in both these expressions we can restrict the summation on τ by only including those (j) permutations which have no fixed points (i.e. α[τ ] ≥ 2 for all j). Such elements define the derangement subset SpD of Sp and by construction can have at most bp/2c cycles, or equivalently τ ∈ SpD. |τ |≥ p − bp/2c ,. =⇒. (38). where |τ | = p − c[τ ] ,. (39). is the minimal number of transpositions (i.e. cycles of length 2) τ is a product of (indeed, if there are more than p/2 cycles in the permutation τ , there must be at least one element i of the set {1, 2, · · · , p} for which τ (i) = i). In the case of Eq. (37) a similar simplification can also be enforced for the summation over σ, as again here one deals with a traceless operator ∆ρ̂. To summarise the following selection rules hold: c[τ ]. τ∈ /. SpD. =⇒. ∆H[τ ] =. Y. (j). Tr[∆Ĥ. α[τ ]. ]=0,. (40). j=1 c[σ]. σ∈ /. SpD. =⇒. ∆ρ[σ] =. Y. (j). α[σ]. Tr[∆ρ̂. ]=0,. (41). j=1. so that we can rewrite Eqs. (36) and (37) as Σ(p) =. X X. C[σ] ρ[στ ] ∆H[τ ]. σ∈Sp τ ∈SpD. =. X. C[στ −1 ] ∆ρ[σ] ∆H[τ ] ,. (42). σ,τ ∈SpD. respectively.. Accepted in. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 9.

(10) p=1:. p=3:. p=4:. C[1] =. C[3] =. 1 d. C[2] = − d(d21−1) ,. p=2:. 2 , (d2 −4)(d2 −1)d. 1 C[2,1] = − (d2 −4)(d 2 −1) ,. 5 C[4] = − (d2 −9)(d2 −4)(d 2 −1)d ,. C[3,1] =. C[1,1,1] =. 2d2 −3 , (d2 −9)(d2 −4)(d2 −1)d2. 2. −4 C[2,1,1] = − (d2 −9)(dd2 −4)(d 2 −1)d ,. C[1,1,1,1] =. 1 d2 −1. ,. d2 −2 (d2 −4)(d2 −1)d. ,. C[1,1] =. C[2,2] =. d2 +6 (d2 −9)(d2 −4)(d2 −1)d2. d4 −8d2 +6 (d2 −9)(d2 −4)(d2 −1)d2. Table 1: List of the first Weingarten functions C[σ] expressed in terms of the dimension d of the system Hilbert space. Data adapted from Ref. [40].. 3.4 Combinatorial coefficients The Weingarten functions C[σ] introduced in Eq. (21) play a fundamental role in the analysis of the moments (42). These terms are in general difficult to compute but a formal expression for them is provided by the formula [41, 42] C[σ] =. 1 X c1 [σ]d2λ , p!2 λ`p sλ,p (1p ). (43). c[λ]≤d. where the sum runs over all the irreducible representations λ of the permutation group Sp which are generated by partitions of d which have at most p elements, dλ is the dimension of the representation λ, c1 [σ] is the number of cycles of length 1 of σ [43] (a quantity sometimes called the character of σ), are the characters, and sλ,p (x1 , . . . , xp ) are the Schur polynomials in p variables (so that sλ,p (1p ) ≡ sλ,p (1, . . . , 1) is the dimension of the representation of the unitary group U(d) which corresponds to λ via the Schur-Weyl duality). For p small a list of the values of the C[σ] is reported in Tab. 1. As the number of irreducible representations of Sp increases with p, the exact computation of these factor becomes rapidly a computationally infeasible task. For the purposes of the present work, however we do not need to compute the coefficients, indeed we just need to use the following known properties. First of all, since C[σ] is a functional of the conjugacy class, exploiting Eq. (26) we can write C[τ στ −1 ] = C[σ] ,. (44). for all τ, σ ∈ Sp . Second, the overall sign of C[σ] is the sign of the permutation σ, namely Sign[C[σ] ] = (−1)|σ| = (−1)p−c[σ] ,. (45). where |σ| is the minimal number of transpositions σ is a product of, see Eq. (39). Finally we shall use the fact that for large enough d the following asymptotic behaviour holds true C[σ] 1 |C[σ] | = p+|σ| + O p+|σ|+2 d d . Accepted in. . ,. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. (46). 10. .. ,.

(11) where the integer number C[σ] – sometimes called the Möbius function of the permutation σ – is equal to the product c[σ]. C[σ] :=. Y. Catα(i) ,. (47). [σ]. i=1 (j). with the α[σ] ’s defined by the correspondence (24), and with 2n 1 Catn := n+1 n. !. ,. (48). being the n-th Catalan number. In particular for the identical permutation this implies 1 1 |C[1p ] | = p + O p+2 d d . . .. (49). The scaling (46) can be derived e.g. from Ref. [44] where Collins and Matsumoto proved that when k j √ (50) p ≤ (d/ 6)4/7 , one has C[σ] p−1 1− 2 d dp+|σ| . −1. C[σ] ≤ |C[σ] | ≤ p+|σ| d. 6p7/2 1− 2 d. !−1. ,. (51). which via some simple algebraic manipulation can be casted in a weaker, but sometimes more useful form: C[σ] p/2 − 1 1+ d2 (d2 − 1)p/2 d|σ| . . C[σ] ≤ |C[σ] | ≤ 2 (d − 1)p/2 d|σ|. 6p7/2 1− 2 d. !−1. .. (52). 4 Bounding the trace terms Here we establish some useful relations that allow us to bound the terms ∆H[τ ] and ∆ρ[σ] entering in Eqs. (42) and (42) and which will allow us to identify the necessary conditions on Ĥ that are needed to prove the Gaussian approximation (10). For this purpose we shall relay on the properties of the derangement set SpD and on the inequality [45] |Tr[Θ̂q2 ]|1/q2 ≤ kΘ̂kq2 ≤ kΘ̂kq1 ,. ∀q2 ≥ q1 > 0 ,. (53). where for q > 0 kΘ̂kq := Tr[|Θ̂|q ]1/q ,. (54). is the q-th Shatten norm of the operator Θ̂.. 4.1 A useful inequality To begin with let consider the special subset SpD∗ of SpD formed by those derangements τ which can be decomposed into p/2 cycles of length 2. If p is even there are (p − 1)!! of such elements, i.e. |SpD∗ | = (p − 1)!! , Accepted in. (p even) .. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. (55) 11.

(12) On the contrary if p is odd there are no such permutations (at least one cycle must be of odd length): in this case we identify SpD∗ with the empty set SpD∗ = ∅ ,. (p odd) .. (56) (j). By definition the elements of SpD∗ verify the condition α[τ ] = 2 for all j and thus, by explicit computation, they fulfil the identity Θ[τ ] = (Tr[Θ̂2 ])p/2 = kΘ̂kp2 ,. ∀τ ∈ SpD∗ ,. (57). for all operators Θ̂. Now let τ ∈ SpD /SpD∗ a derangement which cannot be decomposed into p/2 cycles of length 2: for such permutations we can prove that kΘ̂kp2 provides an upper bound for the associated value |Θ[τ ]|, i.e.. |Θ[τ ]| ≤. kΘ̂kp2. ×.    ηΘ̂ ,. (p even) ,.   √η , Θ̂. (p odd) ,. (58). where ηΘ̂ :=. kΘ̂k3. !6. =. kΘ̂k2. (Tr[|Θ̂|3 ])2 (Tr[Θ̂2 ])3. ,. (59). is a functional of Θ̂ which due to (53) fulfils the inequality [46] 1 ≤ ηΘ̂ ≤ 1 . d6. (60). Equation (58) is a direct consequence of the following more general observation: Proposition 1. Given τ an element of the derangement set SpD we have |τ |−p/2. |Θ[τ ]| ≤ ηΘ̂. |τ |−p/2. (Tr[Θ̂2 ])p/2 = ηΘ̂. kΘ̂kp2 ,. (61). with ηΘ̂ the parameter defined in Eq. (59). Proof. The bound (61) is clearly fulfilled by τ ∈ SpD∗ due to Eq. (57) and the fact that for such permutations c[τ ] = p/2 = |τ |, e.g. see Eq. (39). Now let τ ∈ SpD /SpD∗ a derangement which cannot be decomposed into p/2 cycles (1). (2). (c[τ ]). of length 2. Since τ is an element of SpD all its coefficients {α[τ ] , α[τ ] , . . . , α[τ ] } are guaranteed to be greater or equal to 2, i.e. (j). τ ∈ SpD =⇒ α[τ ] ≥ 2. ∀j .. (62). However since τ ∈ / SpD∗ then some those terms (say the first K ≥ 1) must be strictly larger than 2, i.e. (j). α[τ ] ≥ 3,. ∀1 ≤ j ≤ K .. (63). Invoking (53) we can hence claim (j). α[τ ]. Tr[Θ̂ Accepted in. (j). α[τ ] /3. ] ≤ (Tr[|Θ̂|3 ]). α. (j). = kΘ̂k3 [τ ] ,. ∀1 < j ≤ K ,. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. (64) 12.

(13) and (j). α[τ ]. Tr[Θ̂. ] ≤ Tr[Θ̂2 ]. (j). α[τ ] /2. α. (j). = kΘ̂k2 [τ ] ,. ∀K < j ≤ c[τ ] .. (65). Observe also that since K can be expressed as the number of cycles c[τ ] minus the number c2 [τ ] of cycles of length 2, and since the latter is always smaller than or equal to the minimal number |τ | of transpositions τ can be decomposed of, we can bound this quantity as follows K = c[τ ] − c2 [τ ] ≥ c[τ ] − |τ | = p − 2|τ | ,. (66). where in the last identity we used (39). From the above expressions we can hence establish that for all τ ∈ SpD /SpD∗ we have c[τ ]. |Θ[τ ]| =. PK. (j). α[τ ]. Y. Tr[Θ̂. ] ≤ kĤk3. j=1. (j). α[τ ]. PK. p−. kΘ̂k2. j=1. (j). α[τ ]. j=1. PK. = ηΘ̂. (j). j=1. α[τ ] /6. kΘ̂kp2 ,. (67). where we employ the definition (59) and use the normalization condition Eq. (25) to write c[τ ] X. (j) α[τ ]. =p−. K X (j). α[τ ] .. (68). K p ≥ − |τ | . 2 2. (69). j=1. j=K+1. From Eqs. (63) and (66) we have also (j) j=1 α[τ ]. PK. 6. ≥. Therefore we can claim PK. ηΘ̂. j=1 6. α. (j) [τ ]. p. ≤ ηΘ̂2. −|τ |. ,. (70). which replaced into Eq. (67) finally yields Eq. (61). The derivation of (58) from (61) finally proceed by observing that for τ ∈ SpD /SpD∗ the exponent of ηΘ̂ appearing in (61) is always greater or equal to 1 when p even, and greater or equal to 1/2 for p odd, i.e. |τ | − p/2 = p/2 − c[τ ] ≥.    1,. (p even) ,.   1/2 ,. (p odd) .. (71). 4.2 Condition upon the Hamiltonian Setting Θ̂ = ∆Ĥ in Eqs. (57) and (61) of the previous section we get that the contributions ∆H[τ ] entering in Eqs. (42) and (42) are always upper bounded by the quantity k∆Ĥkp2 , i.e. explicitly ∆H[τ ] = k∆Ĥkp2 ,. |∆H[τ ]| ≤ k∆Ĥkp2 ×. ∀τ ∈ SpD∗ ,    η∆Ĥ. (p even).   √η ∆Ĥ. (p odd). (72). ∀τ ∈ SpD /SpD∗ , (73). Accepted in. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 13.

(14) with . η∆Ĥ :=. k∆Ĥk3 k∆Ĥk2. 6. 1 ∈ 6,1 . d . . (74). As we shall see the fundamental ingredient to prove Eq. (10) is to strengthen the inequality (73) to make sure that the contributions to Σ(p) associated with the derangements τ that do not belong to SpD∗ are very much depressed with respect to k∆Ĥkp2 , i.e. to impose the constraint ∆H[τ ]  k∆Ĥkp2 ,. ∀τ ∈ SpD /SpD∗ ,. (75). k∆Ĥk3  k∆Ĥk2 .. (76). or equivalently to have η∆Ĥ  1 ,. ⇐⇒. It should be clear that due to the fact that since η∆Ĥ is always greater than or equal to 1/d6 , Eq. (76) can only be fulfilled when operating with large Hilbert space, i.e. η∆Ĥ  1. =⇒. d1.. (77). Notice however that once the requirement d  1 is met, the regime (76) is easy to achieve, as it can only fail in the special case of Hamiltonians that have few eigenvalues much greater than all the others.. 5 Central moments asymptotic expression for pure input states In this section we present a proof of our main result that applies in the special scenario where the input state of the system is pure, i.e. ρ̂ = |ψi hψ| .. (78). This constraint allows for some simplifications that help in clarifying the derivation. Indeed thanks to the assumption (78) we have Tr[ρ̂n ] = 1 for every n, which leads to the useful identity c[σ]. ρ[σ] =. (i). α[σ]. Y. Tr[ρ̂. ]=1,. ∀σ ∈ Sp .. (79). i=1. Therefore invoking Eq. (42) we can express the central moment Σ(p) of a pure input state, as the product of a numerical factor that only depends on the geometrical properties of derangements, times a contribution that fully captures the dependence upon the system Hamiltonian, i.e. . Σ(p) = . X σ∈Sp.    X   C[σ]   ∆H[τ ] .. (80). τ ∈SpD. This quantity should be compared with the corresponding Gaussian expression (11) with the scaling factor Gp = Accepted in. (p − 1)!! k∆Ĥkp2 , (d(d + 1))p/2. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. (81) 14.

(15) obtained from (12) using the fact that for pure states Eq. (32) gives us Σ(2) =. k∆Ĥk22 . d(d + 1). (82). Under the above premise in the next section we show that Eq. (13) applies with. fĤ (d, p) =.    dp!/ee p(p−2)  + η − 1 , (p even),   2d ∆Ĥ (p−1)!!   dp!/ee  √η ∆Ĥ (p−1)!! ,. (83). (p odd),. with η∆Ĥ the coefficient defined in Eq. (74). It is worth stressing that in this case, the validity of Eq. (13) is not subjected to any constraint on p and d, but of course fĤ (d, p) can become small only if d is much smaller than p.. 5.1 The geometric coefficient Eq (80) A closed expression for the first factor on the r.h.s. of Eq. (80) is provided by the formula X. C[σ] =. σ∈Sp. (d − 1)! D(d, p) = , (p + d − 1)! (d(d + 1))p/2. (84). (d − 1)!dp/2 (d + 1)p/2 , (p + d − 1)!. (85). with D(d, p) :=. being a numerical coefficient that for p ≥ 2 fulfils the inequalities 1 ≥ D(d, p) ≥.  p(p−2)   1 − 2d  . 1−. (p even), (86). (p+1)(p−1) 2d. (p odd).. An explicit derivation of Eq. (84) can be obtained by focusing on the trivial case where the system Hamiltonian is proportional to the identity operator, i.e. Ĥ = E0 1̂ ,. =⇒. ∆Ĥ = 0 .. (87). In this case in agreement with Eq. (15) the terms (80) vanish leading to the following identities 0 = Σ(p) = hE p i − E0p ,. (88). where we used the fact that now µ = E0 . Invoking hence Eqs. (22) and (79) we can then write . E0p. p. = hE i = .   X. σ∈Sp. C[σ]  .  X. H[τ ] .. (89). τ ∈Sp. P. Our next step is to compute τ ∈Sp H[τ ]. To begin with we write the summation over τ by grouping together those permutations which are characterized by the same value of |τ | = k, decomposing Sp into the subsets Sp (k) := {τ ∈ Sp s.t. |τ | = k} , Accepted in. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. (90) 15.

(16) that is X. H[τ ] =. τ ∈Sp. p X. X. H[τ ] .. (91). k=0 τ ∈Sp (k). Then we recall that the number of elements of Sp (k) corresponds to the numberhof permui p [47]. tations having exaclty p − k cycles, i.e. to the Stirling numbers of the first kind p−k Observe next that from Eq. (87) and (39) we have H[τ ] = E0p 1[τ ] = E0p dc[τ ] = E0p dp−|τ | ,. (92). which replaced into (91) gives X. H[τ ] =. E0p. τ ∈Sp. p X. p−k. . d. k=0. p (p + d − 1)! = E0p , p−k (d − 1)! . (93). where we invoked the useful property [47] p X. dp−k. . k=0. p (p + d − 1)! . = (d − 1)! p−k . (94). Equation (84) finally follows by substituting (93) into (89) and performing some trivial simplifications. We conclude by noticing that the bounds (85) on D(d, p) can be established by observing that it can be expressed as      d d+1 d d+1  · · ·  d+p−2 d+p−1  d+2 d+3. D(d, p) =. (p even),.     √d(d+1)    d d+1 d d+1 d+2 d+3 · · · d+p−3 d+p−2 d+p−1. (p odd),. which immediately reveals that it is always smaller than or equal to 1 for all p ≥ 2. The lower bound instead follows by using the inequalities (1 + x)α > 1 + αx and then Q P i (1 − xi ) > 1 − i xi , and noticing that in case of p even we have . D(d, p) ≥. d d d+2 d+2. p/2−1 . Y. =. . 1+. ··· 2i d. . d d d+p−2 d+p−2. −2. ≥. . p/2−1 . Y. 1−. 4i d. . i=0. i=0 p/2−1. ≥ 1−. X. 4i d. =1−. p(p−2) 2d. .. (95). i=0. For p odd we have instead D(d, p) = =. . . d d+1 d+2 d+3. = Γ(d, p +. d d+1 d+2 d+3. . ···. . . ···. . d d+1 d+p−3 d+p−2. d d+1 d+p−3 d+p−2. .  √d(d+1) d+p−1. d d+1 d+p−1 d+p. . √ d+p. d(d+1). 1) √ d+p d(d+1). ≥ Γ(d, p + 1) ≥ 1 −. (p+1)(p−1) 2d. ,. (96). where the first inequality is obtained by invoking (95). Accepted in. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 16.

(17) 5.2 The Hamiltonian coefficient Eq (80) In studying the Hamiltonian contribution to Eq. (80) for p even we split the summation in two parts writing X. X. ∆H[τ ] =. X. ∆H[τ ] +. ∆H[τ ]. τ ∈SpD /SpD∗. τ ∈SpD∗. τ ∈SpD. = (p − 1)!!k∆Ĥkp2 +. X. ∆H[τ ] ,. τ ∈SpD /SpD∗. (97) where we used the identities (72) and (55) to compute the first contribution. The second contribution instead can be bounded by invoking Eq. (73) to write X. ≤ η∆Ĥ k∆Ĥkp2 SpD /SpD∗. ∆H[τ ]. (98). τ ∈SpD /SpD∗. ≤ η∆Ĥ k∆Ĥkp2 (dp!/ee − (p − 1)!!) , where SpD /SpD∗ := SpD − SpD∗ = SpD − (p − 1)!! ,. (99). is the number of permutation in SpD /SpD∗ which we bounded by exploiting the fact that the number of total elements of the derangement set SpD is [48] SpD = b p!/e + 1/2c ≤ dp!/ee .. (100). For p odd instead SpD∗ is the empty set and we get X. ∆H[τ ]. ≤. √. η∆Ĥ k∆Ĥkp2 SpD. √. η∆Ĥ k∆Ĥkp2 dp!/ee .. τ ∈SpD. ≤. (101). 5.3 Derivation of Eq. (83) Exploiting the results of the previous sections we can conclude that for p even the following inequalities apply (p). D(d,p) (d(d+1))p/2. |Σ(p) − ΣG | =. (p). X. ∆H[τ ] − ΣG. τ ∈SpD. = D(d, p). (p−1)!!k∆Ĥkp2 (d(d+1))p/2. = (D(d, p) −. (p) 1)ΣG. (p). P. +. D /S D∗ ∆H[τ ] τ ∈Sp p (d(d+1))p/2. P. (p). − ΣG. ∆H[τ ]. D /S D∗ τ ∈Sp p (d(d+1))p/2. + D(d, p). ≤ |D(d, p) − 1| ΣG +. !. D(d,p) (d(d+1))p/2. X. ∆H[τ ]. τ ∈SpD /SpD∗. ≤ = Accepted in. p(p−2) (p) 2d ΣG. h. p(p−2) 2d. + η∆Ĥ. + η∆Ĥ. . k∆Ĥkp2 (dp!/ee−(p−1)!!) (d(d+1))p/2. dp!/ee (p−1)!!. −1. i. Gp ,. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. (102) 17.

(18) in agreement with identifying the function fĤ (d, p) of Eq. (13) as anticipated in Eq. (83). Similarly for p odd we can write X D(d, p) ∆H[τ ] p/2 (d(d + 1)) τ ∈S D. |Σ(p) | =. (103). p. ≤. √. η∆Ĥ. k∆Ĥkp2. dp!/ee ≤. (d(d + 1))p/2. √. η∆Ĥ. dp!/ee (p−1)!!. Gp ,. which again corresponds to set fĤ (d, p) of Eq. (13) as in Eq. (83).. 6 Asymptotic expression for the central moments for arbitrary (non necessarily pure) input states Here we present the general proof that, for arbitrary (non necessarily pure) states, under the assumption (106) the centred moments of the distribution P (E|ρ̂; Ĥ) are well approximated by the Gaussian relations (11) whose scaling factors (12) can be conveniently expressed as Gp =. (p − 1)!! k∆Ĥkp2 k∆ρ̂kp2 , (d2 − 1)p/2. (104). upon rewriting Eq. (32) in the compact form k∆Ĥk22 k∆ρ̂k22 . d2 − 1 In particular we shall show that under the condition nj√ k j ko √ p ≤ min d , (d/ 6)4/7 , Σ(2) =. (105). (106). the inequalities (13) hold true with the function fĤ (d, p) defined as . fĤ (d, p) := 1 −. 6p7/2 d2. −1 h. 6p7/2 d2. +. p2 Catp d. + η∆Ĥ. . dp!/ee (p−1)!!. . − 1 Catp (1 +. p2 d ). i. , (107). for p even and η∆Ĥ defined as in Eq. (74), and fĤ (d, p) :=. √. . η∆Ĥ 1 −. 6p7/2 d2. −1 . dp!/ee (p−1)!!. −1. . × Catp (1 +. p2 d ). , (108). for p odd. Notice that weaker, but possibly more friendly, expressions can also be obtained by using the following (generous) upper bound for the Catalan numbers [49] Catp < √. 4p , πp3/2. (109). and observing that under the condition p2 < d, provided that 1 − 6d−1/4 > 0 (i.e., d > 64 = 1296) we can also write 6p7/2 1− 2 d. !−1. ≤. 1 , 1 − 6d−1/4. (110). which does not depend on p. Accepted in. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 18.

(19) 6.1 Case p even In Sec. 4.2 we noticed that if p is even the special set SpD∗ is not empty and it is expected to provide the most relevant contributions in the summation over τ that defines Σ(p) . Accordingly we start splitting the summation (42) in two parts, i.e. Σ(p) = Σ(p)∗ + ∆Σ(p) ,. (111). where the first collects all the terms τ associated with the elements τ ∈ SpD∗ , i.e. X. X. Σ(p)∗ :=. C[στ −1 ] ∆ρ[σ] ∆H[τ ]. σ∈SpD τ ∈SpD∗. =. k∆Ĥkp2. X. X. C[στ −1 ] ∆ρ[σ] ,. (112). σ∈SpD τ ∈SpD∗. which we simplified invoking Eq. (57), and where the second contribution includes instead all the remaining derangements τ , i.e. ∆Σ(p) :=. X. X. σ∈SpD. τ ∈SpD /SpD∗. C[στ −1 ] ∆ρ[σ] ∆H[τ ] .. (113). Observe next that since all the permutations τ entering in the sum (112) belong to the same conjugacy class SpD∗ , we have C[στ −1 ] ∆ρ[σ] = C[σ(σ−1 τ?−1 σ1 )] ∆ρ[σ] 1. = C[(σ−1 σ1 )σσ−1 τ?−1 σ1 ] ∆ρ[σ] 1. 1. = C[σ1 σσ−1 τ?−1 ] ∆ρ[σ] = C[σ0 τ?−1 ] ∆ρ[σ1−1 σ 0 σ1 ] 1. = C[σ0 τ?−1 ] ∆ρ[σ 0 ] ,. (114). where in the first identity we invoked (26) to write τ = σ1 τ? σ1−1 where τ? is a fixed element of SpD∗ and σ1 ∈ Sp , where in the forth line we introduced the permutation σ 0 = σ1 σσ1−1 which inherits from σ the property of being a derangement, and where finally in the third and fifth identity we use the fact C[σ] and ∆ρ[σ] are functional of the conjugacy classes. P From the above identity we can conclude that σ∈SpD C[στ −1 ] ρ[σ] is independent from the specific choice of τ ∈ SpD∗ , i.e. X. C[στ −1 ] ∆ρ[σ] =. X. C[στ?−1 ] ∆ρ[σ] ,. ∀τ ∈ SpD∗ .. (115). σ∈SpD. σ∈SpD. Equation (112) can then be simplified as follows Σ(p)∗ = (p − 1)!!k∆Ĥkp2. X. C[στ?−1 ] ∆ρ[σ] .. (116). σ∈SpD. Now we single out from the above summation the term σ = τ? from the rest identifying the contributions Σ Σ. (p)∗ (p)∗. := (p − 1)!!k∆Ĥkp2 C[1p ] ∆ρ[τ? ] , := (p −. 1)!!k∆Ĥkp2. X. (117). C[στ?−1 ] ∆ρ[σ] ,. σ∈SpD /{τ? }. (118) Accepted in. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 19.

(20) where [1p ] is the conjugacy class of the identical permutation. Accordingly Eq. (116) becomes (p)∗. Σ(p)∗ = Σ. + Σ(p)∗ ,. (119). which replaced into Eq. (111) results into the following decomposition of the p-th centred moment of the distribution P (E; ρ̂, Ĥ), (p)∗. Σ(p) = Σ 6.1.1. + Σ(p)∗ + ∆Σ(p) .. (120). Asymptotic behaviour of the leading term (p)∗. Here we analyze the asymptotic behaviour of Σ of Eq. (117) which, as will shall see is the leading term of the centred moment Σ(p) . Indeed given that τ? is by definition an element of SpD∗ from Eq. (57) we get ∆ρ[τ? ] = k∆ρ̂kp2 , while from Eqs. (49) and (45) we get C[1p ] = (p)∗. Σ. '. 1 dp. (121) . +O. . 1 dp+2. which together leads to. (p − 1)!!k∆Ĥkp2 k∆ρ̂kp2 , dp. (122) (p). that in the large d limit scales exactly as Gp of Eq. (104) and hence as ΣG . More precisely, under the hypothesis (50) we can use Eq. (52) to write C[1p ] −. 1. . ≤. (d2 −1)p/2. 1−. 6p7/2 d2. −1. 6p7/2 (d2 −1)p/2 d2. , (123). and hence (p)∗. Σ. (p). = (p − 1)!!k∆Ĥkp2 k∆ρ̂kp2 C[1p ] −. − ΣG. 1 (d2 −1)p/2. ∗. ≤ Gp f (d, p) ,. (124). with ∗. . f (d, p) := 1 − 6.1.2. 6p7/2 d2. −1. 6p7/2 d2. .. (125). Asymptotic behaviour of Σ(p) (discarded σ’s). Here we evaluate the asymptotic behaviour of Σ(p)∗ of Eq. (118). Let’s start observing that |Σ(p)∗ | ≤ (p − 1)!!k∆Ĥkp2. X. C[στ?−1 ] ∆ρ[σ]. σ∈SpD /{τ? }. ≤ (p − 1)!!k∆Ĥkp2 k∆ρ̂kp2. X. C[στ?−1 ]. σ∈SpD /{τ? }. ≤ (p − 1)!!k∆Ĥkp2 k∆ρ̂kp2. X. C[σ] ,. (126). σ∈Sp /{Id}. where in the second inequality we used the results of Sec. 4.1 to claim that |∆ρ[σ]| ≤ k∆ρ̂kp2 Accepted in. ∀σ ∈ SpD ,. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. (127) 20.

(21) while in the third inequality we extended the summation over σ outside the derangement set, while still keeping out the terms which makes στ?−1 the identity permutation Id. Next P step is to provide an estimation of σ∈Sp /{Id} C[σ] . For this purpose let us decompose Sp in terms of the subsets Sp (k) introduced in Eq. (90) and notice that Sp = ∪pk=0 Sp (k)   . {Id} = Sp (0). =⇒ Sp /{Id} = ∪pk=1 Sp (k) .. (128).  . From this it hence follows that X. C[σ]. p X. =. X. C[σ]. (129). k=1 σ∈Sp (k). σ∈Sp /{Id}. . ≤. 1−. 6p7/2 d2. p −1 X. C[σ] dp+k. X. k=1 σ∈Sp (k). ≤. . =. . 1−. 6p7/2 d2. 1−. 6p7/2 d2. p −1 X. Catp dp+k. k=1. −1. Catp dp. . . p p−k. (p+d−1)! dp (d−1)!. . . −1 ,. where in the first inequality we adopted Eq. (51) which holds under the condition (50); in the second we used [50] . p−k Y. max C[σ] = max . σ∈Sp (k). σ∈Sp (k). . Catα(i) −1  ≤ Catp ,. (130). [σ]. i=1. and compute the total number of elements in Sp (k) via the String number of the fist kind h i p p−k [47]; and finally in the last identity we used (94). Replacing (129) and (104) into Eq. (126) we hence get Σ(p)∗. . 6p7/2 d2. −1. . 6p7/2 d2. −1. ≤ Gp 1 − ≤ Gp 1 −. Catp Catp. √ . d2 −1 d. p . (p+d−1)! dp (d−1)!. (p+d−1)! dp (d−1)!. −1. . −1 .. . (131). Observe that, for p constant, in the large d limit the function multiplying Gp on the r.h.s. tends to zero as O(1/d). Indeed for p2 ≤ d,. (132). noticing that (p + d − 1)! ln p d (d − 1)!. = ln ≤. p−1 Y. i=0 p−1 X i=0. i 1+ d. . p−1 X. i = ln 1 + d i=0 . . i p(p − 1) = , d 2d. (133). (x ≤ 1.256) ,. (134). and using the inequality ex − 1 ≤ 2x Accepted in. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 21.

(22) with x = p2 /d ≥ p(p − 1)/d, the rising factorial can be bounded as p2 (p + d − 1)! − 1 ≤ . dp (d − 1)! d. (135). Exploiting this relation Eq. (131) can finally be casted in the form Σ(p)∗ ≤ Gp f ∗ (d, p) , with 6p7/2 d2. . f ∗ (d, p) := 1 −. −1. (136) p2 Catp d. ,. (137). the expression being valid when both the conditions (50) and (132) apply, or in brief when Eq. (106) holds. 6.1.3. Asymptotic behaviour of ∆Σ(p) (discarded τ ’s). Here we study the term ∆Σ(p) which according to Eq. (113) is obtained by running the sum over τ by only including derangements that are not in the special subset SpD∗ . Accordingly invoking Eq. (73) and (127) we can write: |∆Σ(p) | ≤. X. X. σ∈SpD. τ ∈SpD /SpD∗. |C[στ −1 ] | |∆ρ[σ]| |∆H[τ ]|. ≤ η∆Ĥ k∆Ĥkp2 k∆ρ̂kp2 ≤ η∆Ĥ k∆Ĥkp2 k∆ρ̂kp2. X. X. σ∈SpD. τ ∈SpD /SpD∗. X. |C[στ −1 ] |. X. |C[στ −1 ] |. σ∈Sp τ ∈SpD /SpD∗. = η∆Ĥ k∆Ĥkp2 k∆ρ̂kp2. X. X. |C[σ] |. σ∈Sp τ ∈SpD /SpD∗. = η∆Ĥ k∆Ĥkp2 k∆ρ̂kp2 SpD /SpD∗. X. |C[σ] |. σ∈Sp. ≤ η∆Ĥ k∆Ĥkp2 k∆ρ̂kp2 (dp!/ee − (p − 1)!!) 6p7/2 d2. . × 1−. −1. Catp (p+d−1)! dp dp (d−1)!. , (138). where we used (99) and (100) to get an estimate of SpD /SpD∗ and follow the same passages of Eq. (129) to get X. . C[σ] ≤ 1 −. 6p7/2 d2. −1. Catp (p+d−1)! dp dp (d−1)!. .. (139). σ∈Sp. Using hence Eq. (104) we can then translate Eq. (138) into |∆Σ(p) | ≤ η∆Ĥ Gp. . dp!/ee (p−1)!!. × Catp ≤ η∆Ĥ Gp. . dp!/ee (p−1)!!. −1. . 1−. 6p7/2 d2. −1  √. d2 −1 d. p. (p+d−1)! dp (d−1)!. −1. . 1−. 6p7/2 d2. −1. Catp. (p+d−1)! dp (d−1)!. , (140). Accepted in. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 22.

(23) where, using Eq. (135) and assuming (132), the last coefficient can be further bounded from above as (p + d − 1)! p2 ≤1+ , p d (d − 1)! d. (141). that shows that in the large d limit this term converges to 1. To summarize we can hence conclude that |∆Σ(p) | ≤ η∆Ĥ Gp ∆f (d, p) , with ∆f (d, p) :=. . dp!/ee (p−1)!!. . −1. 1−. 6p7/2 d2. −1. (142). Catp (1 +. p2 d ). ,. (143). which again is valid under the condition (106). 6.1.4. Total scaling (p). From Eqs. (124), (136), and (142) we can now estimate the distance of Σ(p) from ΣG : indeed from Eq. (120), we get that under the assumption (106) we can write (p)∗. (p). |Σ(p) − ΣG | ≤ |Σ. (p). − ΣG | + |Σ(p)∗ | + |∆Σ(p) |. ≤ Gp fĤ (d, p) , with. (144). ∗. fĤ (d, p) := f (d, p) + f ∗ (d, p) + η∆Ĥ ∆f (d, p) ,. (145). exactly matching the expression given in Eq. (13).. 6.2 Case p odd The proof of Eq. (13) in the case of p odd is easier to treat as now the set SpD∗ is empty, i.e. a condition which we can summarize by saying that Σ(p)∗ = 0. Accordingly we can treat the whole Σ(p) as we treated the component ∆Σ(p) of the even p case. In particular the entire derivation of Sec. 6.1.3 can be repeated with the minor modification that according √ to (58) the coefficient ηĤ entering into (142) gets replaced by ηĤ . Therefore in this case we have √ |Σ(p) | ≤ ηĤ Gp ∆f (d, p) , (146) with ∆f (d, p) as in Eq. (143), which corresponds to the expression given in Eq. (108).. 7 Moment generating function In this section we shall estimate the distance between the generating function of the moments of the distribution P (E; ρ̂, Ĥ), i.e. D. E. G(t) := et(E−µ) =. ∞ p (p) X t Σ p=0. p!. ,. (µ,Σ(2) ). and the one associated with the Gaussian distribution PG ∞ p (p) X t ΣG p=0. Accepted in. p!. 1 2 (2) Σ. = e2t. (147). (E), i.e.. ,. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. (148). 23.

(24) (p). where ΣG are the Gaussian moments defined in Eq. (10). In particular we shall show that, for values of |t| such that (√ ) N? N? |t| ≤ min √ , , Σ(2) ∆E max where ∆E max := max. n. (149). Tr[ρ̂(↓) Ĥ] − µ , Tr[ρ̂(↑) Ĥ] − µ. o. .. (150). and N ? := min. nj√ k j. ko √ , d , (d/2 3)4/7. (151). then the following inequality holds: 1 2 (2) Σ. 2 Σ(2). G(t) − e 2 t. ≤ e2t 2. (2). Σ + 32t√πd. . h. p. −4|t|. Σ(2) /η.  e. + 2√1 π . 2. (2). p 32 q ηĤ /π · t Σ(2) 3. 3. . 2 Σ(2). e16t. −1. . i Ĥ.  1 √ 1−4|t| ηĤ Σ(2) ηĤ. + dN ?1/2e! 1− tt2 ΣΣ(2) /N ? + ( ) =. . − 2t2 Σ(2) − 1. 2 Σ(2). − e16t.  . |t|∆E max 1 N ? ! 1−(|t|∆E max /N ? ). . + O ηĤ t4. p. Σ(2). 2 . . (152). The error D E (152) is also valid for the characteristic function of the distribution, namely −itE e [51].. 7.1 Initial considerations We shall estimate the maximum distance between the moment generating functions with the series ∞ p X 1 2 (2) t (p) G(t) − e 2 t Σ ≤ Σ(p) − ΣG . (153) p! p=0 All the estimations we made in Sec. 6 are valid only under the assumption (106). At (p) some point, the momenta Σ(p) will become much more different from ΣG , because the (µ,Σ(2) ). distribution P (E|ρ̂; Ĥ) has a compact support, while the gaussian distribution PG (E) has infinite tails which become more and more important for the momenta of higher order. For this reason, it is convenient to consider separately the terms of the sum on the r.h.s. of Eq. (153) with p not fulfilling (106) from the rest writing 1 2 (2) Σ. G(t) − e 2 t. ≤ Γ∗< + ∆Γ< + Γ> ,. (154). where, given the cutoff (151), we set Γ> :=. ∞ X tp p=N ?. Accepted in. p!. (p). Σ(p) − ΣG. ,. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. (155). 24.

(25) and, invoking the decomposition (111), Γ∗< :=. N? p X t. ?. (p). Σ(p)∗ − ΣG. p! p=1. N X tp. =. p=4 p even. (p). Σ(p)∗ − ΣG. p!. , (156). ∆Γ< :=. N? p X t. ∆Σ(p) ,. p! p=1. (157). with Σ(p)∗ and ∆Σ(p) ) defined as in Eqs. (112) and (113) – notice that Eq. (156) was (p) simplified by using the fact that for p odd both Σ(p)∗ and ΣG are equal to zero, while by (2) definition Σ(2)∗ = ΣG .. 7.2 Bound on Γ> When p is large enough the condition (106) brakes and we can not use anymore the bounds of Sec. 6. However since the spectrum of the random variable E is limited as in Eq. (14), for all p we can write Σ(p) ≤ ∆E pmax ,. (158). with ∆E pmax as defined in (150). Therefore we can write (p). Σ(p) − ΣG. = Σ(p) ≤ ∆E pmax ,. (159). for p odd, and (p). Σ(p) − ΣG. (p). (p). ≤ Σ(p) + ΣG ≤ ∆E pmax + ΣG ,. (160). for p even. Exploiting these inequalities, when (149) holds we can estimate the quantity (155) as. Γ> ≤.  p/2 ∞ tp (p − 1)!! Σ(2) X p=N ? p even. p!. ∞ X tp ∆E pmax. +. p=N ? ∞ X. t2 Σ(2) 2. 1 ≤ n! n=dN ? /2e ?. ≤. p!. (N ? )dN /2e dN ? /2e!. !n. +. +. N ?!. p!. p=N ?. ∞ X n=dN ? /2e. ? (N ? )N. ∞ X tp ∆E pmax. t2 Σ(2) N?. !n.  ∞  X t∆E max p p=N ?. N?. 1 t2 Σ(2)  dN ? /2e! 1 − t2 Σ(2) /N ? t∆E max 1 + ? N ! 1 − (t∆E max /N ? ). ≤. 1 =O ? dN /2e! . Accepted in. . . = O  l√. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 1.  m  . (161). d/2 !. 25.

(26) 7.3 Bound on Γ∗< From Eq. (119) we can write ∗. Γ∗< ≤ Γ< + Γ∗< ,. (162). where ?. ∗ Γ<. N X tp. :=. p=4 p even. p!. Σ. (p)∗. (p). − ΣG. ,. (163). ?. Γ∗<. N X tp. :=. p=4 p even. p!. |Σ(p)∗ | ,. (164). (p)∗. and Σ(p)∗ as defined in (117) and (118). and with Σ ∗ First we provide a bound for the term Γ< . Retrieving from (124) the bound on Σ. (p)∗. (p). − ΣG , and using the expression (105) for the variance Σ(2) , we have ∗ Γ<. ≤. ≤. ∞  X (p−1)!! p! p=4 p even ∞ X p=4 p even. 1−. 6p7/2 d2. −1. 6p7/2 tp d2. . Σ(2). p/2. ∞ 2n t2n  (2) n (p − 1)!! p  (2) p/2 X ≤ t Σ Σ p! n! n=2. h. i. = exp 2t2 Σ(2) − 2t2 Σ(2) − 1 . = O t. 4. . (2). Σ. 2 . ,. (165).  √ 4/7 in the second inequality we used the fact that the condition p ≤ N ? ≤ d/2 3 implies 6p7/2 d2. . 1−. 6p7/2 d2. −1. ≤1.. (166). Regarding Γ∗< , recalling equation (136), and expliciting the definitions (137) and (104) of the quantities f ∗ (d, p) and Gp we have that Σ(p)∗ ≤. (p−1)!! k∆Ĥkp2 (d2 −1)p/2. . k∆ρ̂kp2 1 − . = (p − 1)!! 1 −. 6p7/2 d2. 6p7/2 d2. −1. −1 . p2 Catp d. Σ(2). ,. p/2. (167). Accepted in. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 26.

(27) from which, using the bound (109) on Catalan numbers we can write Γ∗<. N?  X (p−1)!!. ≤. 1−. 6p7/2 d2. −1. d X (2n − 1)!! . 2. (2). p!. √ (4t)p p √ πd. p=4 p even. ≤ 2 ≤ 2. (2n)!. n=2 ∞ X. 16t Σ. n. . Σ(2). p/2. √ 2n √ πd. (2n − 1)!!  2 (2) n n √ 16t Σ (2n)! πd n=2. =. ∞ 1 X 2n  2 (2) n √ 16t Σ πd n=2 n!. =. ∞ 1 X t ∂  2 (2) n √ 16t Σ πd n=2 n! ∂t. =. ∞ t ∂ X 1  2 (2) n √ 16t Σ πd ∂t n=2 n!. =.   i t ∂ h √ exp 16t2 Σ(2) − 16t2 Σ(2) − 1 πd ∂t. =.   i 32t2 Σ(2) h √ exp 16t2 Σ(2) − 1 , πd . . = O t4 Σ(2). 2. d−1. . , (168). where in the second inequality we let n = p/2 and used the fact that, since p ≤ N ? ≤  √ 4/7 d/2 3 , . 7/2. −1. 1 − 6pd2 √ and in the third inequality we used 2n ≤ n.. ≤2,. (169). 7.4 Bound on ∆Γ< From the definitions (157) and (113), we can write the term ∆Γ< as ∆Γ< =. ∞ p X X t p=1. p!. X. C[στ −1 ] ∆ρ[σ] ∆H[τ ] .. σ∈SpD τ ∈SpD /SpD∗. (170). Accepted in. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 27.

(28) First we use (130), as well as ∆ρ[στ ] ≤ k∆ρ̂kp2 , to write |∆Γ< | ≤. <. ∞ p X t. p! p=3. . k∆ρkp2 . C[σ]  .  X. p!. ∞ p  X t p=3. p!. 1−. 1−. 6p7/2 d2.  X. . ∆H[τ ] . τ ∈SpD /SpD∗. σ∈SpD. ∞ p X t k∆ρkp2  p=3. =. . −1. Catp (d+p−1)! d2p (d−1)!. X. H[τ ]. τ ∈SpD /SpD∗ 6p7/2 d2. −1. p X. Catp (d+p−1)! d2p (d−1)!. X. H[τ ] ,. k=bp/2c+1 τ ∈SpD (k). (171) where in thethird line we introduced the sets SpD (k) := SpD ∩ Sp (k) ,. (172). and noticed that SpD (k) = ∅ when k < bp/2c + 1 and that, when p is even, SpD (p/2) = SpD∗ (recall also that SpD∗ = ∅ in the case of odd p). Replacing the bounds (109) and (135) in (171), we have |∆Γ< | ≤ p2 × 1+ d. !. ∞ X 4p tp k∆ρ̂kp2 . √. p=3 p X. πdp p3/2 p!. 1−. k−p/2. X. k=bp/2c+1 τ ∈SpD (k). ηĤ. 6p7/2 d2. −1. k∆Ĥkp2 .. (173). Now, in order to √ simplify the calculations, we will use the inequality (169) to infer that, since 4 ≤ p ≤ d, p. −3/2. 6p7/2 1− 2 d. !−1. p2 1+ d. !. < 4p−3/2 ≤. 1 . 2. With the help of (174), and noticing from (105) that k∆Ĥk2 k∆ρ̂k2 /d < turn (173) into. (174) √. Σ(2) , we can.   ∞ 1 X 1 4tk∆Ĥk2 k∆ρ̂k2 p |∆Γ< | ≤ √ d 2 π p=3 p! p X. ×. X. k=bp/2c+1 τ ∈SpD (k) p ∞ 1 X 1  p (2) p X ≤ √ 4t Σ 2 π p=3 p! k=bp/2c+1. 1 = √ 2 π.  q ∞ X 1. p! p=3. 4t ηĤ Σ(2). p. X τ ∈SpD (k). p X. k−p/2. ηĤ. k−p/2. ηĤ. k−p , dp,k ηĤ. k=bp/2c+1. (175) where we called dp,n the number of elements in the set SpD (k).. Accepted in. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 28.

(29) We do not need the explicit values of dp,k . They are implicitly given by the coefficients in the expansion of the exponential generating function[52] ∞ X ∞ X. dp,p−i ci. i=1 p=i. e−cx xp = . p! (1 − x)c. (176). In wiev of (175), we want to keep in the sum only the derangements with less than p/2 cycles (τ 6∈ [2p/2 ]), whose generating functions is ∞ X. p X. dp,k cp−k. p=2 k=bp/2c−1. ∞ dp/2e−1 X X xp p = dp,p−i ci xp! p! p=2 i=1. =. ∞ bp/2c X X. p ix. dp,p−i c. p=0 i=1. p!. ∞ X (p−1)!! p/2 p − x p! c p even. =.   cx2 cx3 e−cx 4 2 = − e + O cx . (177) (1 − x)c 3 q. −1 , we can Using (177) into (175), with the identifications x = 4t ηĤ Σ(2) and c = ηĤ conclude that. . 1 |∆Γ< | ≤ √ 2 π =. . . √. . (2)   e −4|t| Σ /ηĤ  16t2 Σ(2)  − e   q  1  . 1 − 4|t| ηĤ Σ(2). p 32 q ηĤ /π · t Σ(2) 3. 3. . ηĤ. + O ηĤ t4. p. Σ(2). 2 . . (178). 8 Conclusions We derived a bound for the distance between the energy distribution in the unitary orbit of a quantum state and the normal distribution with the same variance, showing that for large dimensions of the Hilbert space the difference in the charachteristic function is suppressed as d−3 . We have also characterised the individual moments of the distribution: these result apply also to unitary t-design, which mimic the uniform distribution of unitary transformations up to the t-th moments. Our findings, therefore, are suitable to be be employed for certifying that an unitary t-design behaves as expected. Since the results do not depend on the specific form of the Hamiltonian (but only on a very general hypotesis on its eigenvalues), they can be applied to every observable that can be measured on the system. The validity of our results is limited to the specific (but relevant in the field of quantum computing) case in which the evolution of the system is described by a random unitary matrices drawn from the Haar-uniform ensemble (also known as Circular Unitary Ensemble [53, 54]). In our opinion, it would be interesting to perform a similar analysis on the work distribution in the case in which the time evolution of the system is given by an Hamiltonian drawn from the Wigner-Dyson Gaussian Unitary Ensemble [55, 56], which describes quantum chaotic systems lacking time-reversal symmetry [57, 58], or from the Gaussian Orthogonal or the Gaussian Symplectic Ensembles, which model the evolution quantum chaotic systems in presence time-reversal symmetry [58]. The combinatorical Accepted in. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 29.

(30) tools necessary for this analogous task have been in part already developed [59], but significant more work may be required. Another possible future developement of this work is generalizing it to Hilbert space of infinite dimension. An immediate difficulty is that the Haar measure on the unitary group of infinite dimension is not well-defined [60]. However, a class of states of particular interest are the gaussian states on systems charachterised by a N -modes quadratic bosonic Hamiltonian[61]: in this setting, it could be of some interest to study the distribution of their energy over the group of symplectic transformations [62]. The Authors acknowledge support from PRIN 2017 (Progetto di Ricerca di Interesse Nazionale): project “Taming complexity with quantum strategies” QUSHIP (2017SRNBRK).. References [1] A. Hayashi, T. Hashimoto, and M. Horibe. Reexamination of optimal quantum state estimation of pure states. Physical Review A, 72(3), September 2005. DOI: 10.1103/physreva.72.032325. URL https://doi.org/10.1103/physreva.72. 032325. [2] A J Scott. Optimizing quantum process tomography with unitary2-designs. Journal of Physics A: Mathematical and Theoretical, 41(5):055308, January 2008. DOI: 10.1088/1751-8113/41/5/055308. URL https://doi.org/10.1088/1751-8113/41/ 5/055308. [3] Fernando G. S. L. Brandão and Michal Horodecki. Exponential quantum speed-ups are generic. Quantum Info. Comput., 13(11–12):901–924, November 2013. ISSN 1533-7146. [4] Joseph Emerson, Robert Alicki, and Karol Życzkowski. Scalable noise estimation with random unitary operators. Journal of Optics B: Quantum and Semiclassical Optics, 7(10):S347–S352, September 2005. DOI: 10.1088/1464-4266/7/10/021. URL https://doi.org/10.1088/1464-4266/7/10/021. [5] David C. McKay, Sarah Sheldon, John A. Smolin, Jerry M. Chow, and Jay M. Gambetta. Three-qubit randomized benchmarking. Physical Review Letters, 122(20), May 2019. DOI: 10.1103/physrevlett.122.200502. URL https://doi.org/10.1103/ physrevlett.122.200502. [6] Jonas Helsen, Xiao Xue, Lieven M. K. Vandersypen, and Stephanie Wehner. A new class of efficient randomized benchmarking protocols. npj Quantum Information, 5 (1), August 2019. DOI: 10.1038/s41534-019-0182-7. URL https://doi.org/10. 1038/s41534-019-0182-7. [7] D.P. DiVincenzo, D.W. Leung, and B.M. Terhal. Quantum data hiding. IEEE Transactions on Information Theory, 48(3):580–598, March 2002. DOI: 10.1109/18.985948. URL https://doi.org/10.1109/18.985948. [8] Adam Nahum, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah. Quantum entanglement growth under random unitary dynamics. Phys. Rev. X, 7:031016, Jul 2017. DOI: 10.1103/PhysRevX.7.031016. URL https://link.aps.org/doi/10. 1103/PhysRevX.7.031016. [9] Cheryne Jonay, David A. Huse, and Adam Nahum. Coarse-grained dynamics of operator and state entanglement. 2018. [10] Daniel Gottesman. Theory of fault-tolerant quantum computation. Physical Review A, 57(1):127–137, January 1998. DOI: 10.1103/physreva.57.127. URL https://doi. org/10.1103/physreva.57.127. Accepted in. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 30.

(31) [11] Victor Veitch, S A Hamed Mousavian, Daniel Gottesman, and Joseph Emerson. The resource theory of stabilizer quantum computation. New Journal of Physics, 16(1): 013009, January 2014. DOI: 10.1088/1367-2630/16/1/013009. URL https://doi. org/10.1088/1367-2630/16/1/013009. [12] D. Gross, K. Audenaert, and J. Eisert. Evenly distributed unitaries: On the structure of unitary designs. Journal of Mathematical Physics, 48(5):052104, May 2007. DOI: 10.1063/1.2716992. URL https://doi.org/10.1063/1.2716992. [13] Adam Sawicki and Katarzyna Karnas. Universality of single-qudit gates. Annales Henri Poincaré, 18(11):3515–3552, August 2017. DOI: 10.1007/s00023-017-0604-z. URL https://doi.org/10.1007/s00023-017-0604-z. [14] Eiichi Bannai, Gabriel Navarro, Noelia Rizo, and Pham Huu Tiep. Unitary t-groups. arXiv [math.RT], Oct 2018. URL https://arxiv.org/abs/1810.02507. [15] Jonas Haferkamp, Felipe Montealegre-Mora, Markus Heinrich, Jens Eisert, David Gross, and Ingo Roth. Quantum homeopathy works: Efficient unitary designs with a system-size independent number of non-clifford gates. arXiv [quant-ph], Feb 2020. URL https://arxiv.org/abs/2002.09524. [16] Massimiliano Esposito, Upendra Harbola, and Shaul Mukamel. Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Reviews of Modern Physics, 81(4):1665–1702, December 2009. DOI: 10.1103/revmodphys.81.1665. URL https://doi.org/10.1103/revmodphys.81.1665. [17] P. Talkner M. Campisi, P. Hänggi. Colloquium. quantum fluctuation relations: Foundations and applications. Rev. Mod. Phys., 83:771–791, 2011. DOI: 10.1103/RevModPhys.83.771. [18] Thomas Speck and Udo Seifert. Distribution of work in isothermal nonequilibrium processes. Physical Review E, 70(6), December 2004. DOI: 10.1103/physreve.70.066112. URL https://doi.org/10.1103/physreve.70.066112. [19] Evžen Šubrt and Petr Chvosta. Exact analysis of work fluctuations in two-level systems. Journal of Statistical Mechanics: Theory and Experiment, 2007(09):P09019– P09019, September 2007. DOI: 10.1088/1742-5468/2007/09/p09019. URL https: //doi.org/10.1088/1742-5468/2007/09/p09019. [20] Thomas Speck. Work distribution for the driven harmonic oscillator with timedependent strength: exact solution and slow driving. Journal of Physics A: Mathematical and Theoretical, 44(30):305001, June 2011. DOI: 10.1088/17518113/44/30/305001. URL https://doi.org/10.1088/1751-8113/44/30/305001. [21] Harry J. D. Miller, Matteo Scandi, Janet Anders, and Martı́ Perarnau-Llobet. Work fluctuations in slow processes: Quantum signatures and optimal control. Physical Review Letters, 123(23), December 2019. DOI: 10.1103/physrevlett.123.230603. URL https://doi.org/10.1103/physrevlett.123.230603. [22] Matteo Scandi, Harry J. D. Miller, Janet Anders, and Martı́ Perarnau-Llobet. Quantum work statistics close to equilibrium. Phys. Rev. Research, 2:023377, Jun 2020. DOI: 10.1103/PhysRevResearch.2.023377. URL https://link.aps.org/doi/10. 1103/PhysRevResearch.2.023377. [23] C. Jarzynski. Nonequilibrium equality for free energy differences. Physical Review Letters, 78(14):2690–2693, April 1997. DOI: 10.1103/physrevlett.78.2690. URL https://doi.org/10.1103/physrevlett.78.2690. [24] W. Pusz and S. L. Woronowicz. Passive states and KMS states for general quantum systems. Communications in Mathematical Physics, 58(3):273–290, October 1978. DOI: 10.1007/bf01614224. URL https://doi.org/10.1007/bf01614224.. Accepted in. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 31.

(32) [25] A. Lenard. Thermodynamical proof of the gibbs formula for elementary quantum systems. Journal of Statistical Physics, 19(6):575–586, December 1978. DOI: 10.1007/bf01011769. URL https://doi.org/10.1007/bf01011769. [26] A. E. Allahverdyan, R Balian, and Th. M. Nieuwenhuizen. Maximal work extraction from finite quantum systems. Europhysics Letters (EPL), 67(4):565–571, aug 2004. DOI: 10.1209/epl/i2004-10101-2. URL https://doi.org/10.1209%2Fepl% 2Fi2004-10101-2. [27] Francesco Campaioli, Felix A. Pollock, and Sai Vinjanampathy. Quantum batteries. In Fundamental Theories of Physics, pages 207–225. Springer International Publishing, 2018. DOI: 10.1007/978-3-319-99046-0˙8. URL https://doi.org/10.1007/ 978-3-319-99046-0_8. [28] R. Alicki and M. Fannes. Entanglement boost for extractable work from ensembles of quantum batteries. Phys. Rev. E, 87:042123, 2013. [29] Karen V. Hovhannisyan, Martı́ Perarnau-Llobet, Marcus Huber, and Antonio Acı́n. Entanglement generation is not necessary for optimal work extraction. Physical Review Letters, 111(24), dec 2013. DOI: 10.1103/physrevlett.111.240401. URL https://doi.org/10.1103/physrevlett.111.240401. [30] Felix C Binder, Sai Vinjanampathy, Kavan Modi, and John Goold. Quantacell: powerful charging of quantum batteries. New Journal of Physics, 17(7):075015, July 2015. DOI: 10.1088/1367-2630/17/7/075015. URL https://doi.org/10.1088/ 1367-2630/17/7/075015. [31] Sergi Julià-Farré, Tymoteusz Salamon, Arnau Riera, Manabendra N. Bera, and Maciej Lewenstein. Bounds on the capacity and power of quantum batteries. Physical Review Research, 2(2), May 2020. DOI: 10.1103/physrevresearch.2.023113. URL https://doi.org/10.1103/physrevresearch.2.023113. [32] . Given f (E) a generic function of the random variable (8), its mean value with respect to P (E|ρ̂; Ĥ), i.e. the quantity hf (E)i :=. Z. dEP (E|ρ̂; Ĥ)f (E) Z. dµ(Û )f (E(Û ρ̂Û † ; Ĥ)),. =. (179). with dµ(Û ) representing the Harr measure on U(d). [33] . Notice that the quantity (20) can be equivalently be expressed as h. hE p i = Tr T (p) (ρ̂⊗p )Ĥ ⊗p where T (p) (· · · ) :=. Z. i. dµ(Û ) Û ⊗p · · · Û †⊗p. are Completely Positive, trace preserving map known as Twirling channels which have a number of applications in quantum information theory [63–68]. [34] Don Weingarten. Asymptotic behavior of group integrals in the limit of infinite rank. Journal of Mathematical Physics, 19(5):999–1001, May 1978. DOI: 10.1063/1.523807. URL https://doi.org/10.1063/1.523807. [35] Michael Creutz. On invariant integration over SU(n). Journal of Mathematical Physics, 19(10):2043, 1978. DOI: 10.1063/1.523581. URL https://doi.org/10. 1063/1.523581. [36] I. Bars. U(n) integral for the generating functional in lattice gauge theory. Journal of Mathematical Physics, 21(11):2678–2681, November 1980. DOI: 10.1063/1.524368. URL https://doi.org/10.1063/1.524368. Accepted in. Quantum 2021-07-20, click title to verify. Published under CC-BY 4.0.. 32.

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