Quantum Chaos is Quantum

It is well known that a quantum circuit on $N$ qubits composed of Clifford gates with the addition of $k$ non Clifford gates can be simulated on a classical computer by an algorithm scaling as $\text{poly}(N)\exp(k)$[1]. We show that, for a quantum circuit to simulate quantum chaotic behavior, it is both necessary and sufficient that $k=O(N)$. This result implies the impossibility of simulating quantum chaos on a classical computer.


Introduction
Quantum chaos is a certain type of complex quantum behavior that results in the exponential decay of outof-time-order correlation functions (OTOC) [2][3][4] efficient operator spreading [5,6], small fluctuations of the purity [7] and information scrambling [8,9]. All these quantities can be unified in a single framework [10] which shows that, in order to simulate quantum chaos, one needs at least a unitary 4-design, that is, a set of unitary operators that reproduces up to the four moments of the Haar distribution over the unitary group U(d) in an d-dimensional Hilbert space. In [11] it was shown numerically that a Clifford circuit on a d = 2 N -dimensional system of N qubits doped by a single T gate can bring a typical product state in an entangled state with the same entanglement spectrum statistics resulting from the random matrix theory for U(d). This result opens the question whether it would be possible to simulate quantum chaos by classical resources. In a seminal paper [12], the authors show that an -approximate t-design can be obtained by doping a Clifford circuit with k = O(t 4 log 2 t log −1 ) non Clifford gates. In particular, one can -simulate the quantum channel that realizes a 4-design by classical resources. This result is striking: by injecting a vanishing density σ = k/N of non Clifford gates in a Clifford quantum circuit -as the authors say, homeopathically -one can obtain any -approximate t-design. Does this mean that one can simulate quantum chaos classically? The answer is no, because -as we will show -to simulate quantum chaos, the error must be exponentially small in N , = O(d −α ), where α only depends on the Haar average over the full unitary group. A corollary of the result in [12] is that a sufficient condition to simulate quantum chaos requires O(N ) non Clifford resources.
In this paper, we show that O(N ) non Clifford resources are both necessary and sufficient to simulate quantum chaos. To this end, we explicitly compute the 8-point OTOC and the fluctuations of the purity in a subsystem and show that a doped Clifford circuit will attain the Haar values for these quantities if and only if O(N ) non Clifford resources are used. In other words, one needs more than an homeopathic dose of non Clifford gates to simulate quantum chaos and therefore quantum chaos cannot be efficiently simulated on a classical computer: quantum chaos is quantum.

Doped Random Quantum Clifford Circuits
Consider doped random quantum Clifford circuits U (k) on a system H = C 2⊗N of N qubits of dimension d = 2 N . The architecture of the circuit is the following: we have layers of random Clifford unitary operators on the full H interspersed by a single qubit gate K i applied randomly on any qubit i, see Fig.1. As we shall see in Sec. C.2, the positioning i of the gates K does not in fact play any role. We denote by k the number of gates K in the circuit, also called the number of layers of the circuit, ψ a pure input state for the circuit, and ψ U = U ψU † its output. We call the quantity σ = k/N the doping of the circuit U (k) . We denote by x a set of unitary operators, e.g. x = U(d), C(d) the unitary and Clifford group, respectively, on H. For k = 0, the circuit is just a Clifford circuit, U (0) ∈ C(d). The Haar average on these sets will be denoted by · U ∈x . We define the (x, t)-fold channel as where O ∈ B(H ⊗t ) and O U ≡ U ⊗t OU †⊗t . Averaging over C(d) for a circuit U (k) with k layers involves averaging over k independent Clifford groups; in the following we define this set of circuits as C k . The (2) Notice that the above average over C(d) is the same thing -because of the left/right invariance of group averages -than the average over circuits of the type sketched in Fig.1.
Quantum chaos can be defined as an appropriate form of the butterfly effect [13]: an exponential (in N ) decay of the OTOCs defined as so that the OTOCs adhere to the value of the OTOCs obtained by Haar-random U on the unitary group scaling with d −4 , while other ensembles, like the Clifford group, feature a scaling of d −2 [3]. It is immediate to see that, in order to distinguish the two types of scaling, one needs an = O(d −4 ). As 2t-OTOCs are probes of t-designs, a 8-point OTOC is a probe of a 4-design and therefore a quantum chaotic channel needs to have a frame potential exponentially close to that of the Haar measure on U(d).
A related measure of chaos [10] is given by the fluctuations of the purity of the reduced density matrix to a subsystem This quantity is related to the emergent irreversibility in closed quantum systems [14] and to both 4-designs and OTOCs. In Sec. 4, we show that the purity fluctuations are exponentially small for every doping (including no doping, k = 0) of the random Clifford circuits and thus also to distinguish the fluctuations of the purity one needs an exponentially small error . We ask the question: what is the necessary and sufficient number k of non Clifford gates K for U ∈ C k to simulate quantum chaos?
The main goal of this paper is to show that, for C k to reproduce the Haar-unitary values of the probes Eqs. (3) and (4), O(N ) non Clifford resources are both necessary and sufficient. We will prove it in the next sections by explicitly computing these two quantities. Here, we want to make some more general considerations. Given a probe to quantum chaos defined as P t (U ) := tr(T (t) O 1 U ⊗t O 2 U †⊗t ), see [7,10], we can establish the following Proposition 1 Let P t (U ) a probe of quantum chaos of order t. If the number k of non Clifford gates in the doped Clifford circuit U ∈ C k is k = O((α + t)N t 4 log 2 (t)), then: where α is given by the following relation Proof.-The proof is straightforward from the result in [12]. In [10], we proved that the generic probe P t (U ) to quantum chaos can be written as P t (U ) = tr( , including the 2t-point OTOC which characterize t-designs [3]. Then, the following inequality holds: 2 As we will show in the following sections, the error O(d −α ) is required to have OTOCs and fluctuations of the purity attain the unitary-Haar values. It follows that injecting O(N ) non Clifford resources into a Clifford circuit is sufficient to obtain quantum chaos.
As we stated above, the necessary (together with the sufficient) condition will follow from direct calculation. It is important at this point to make some remarks about the value of t. One wonders if it is enough to consider 8-OTOCs to reveal quantum chaos, or if sometimes it should be necessary to use higher order OTOCs. From the point of view of the above proposition, it is clear that O(N ) non Clifford resources are sufficient to obtain any OTOC with an exponentially good O(d −α ) approximation. The necessary condition holds trivially for every t > 4 design, as an approximate t-design is necessarily a t approximate design, for any t < t. In other words, polynomials of degree four is all that takes to reveal quantum chaos.

Main Theorem
From the technical point of view, the main result of this paper is the exact calculation of the fourth moment of the output of a k-doped random Clifford circuit for a generic operator O ∈ B(H ⊗4 ): Theorem 1 Let O ∈ B(H ⊗4 ) be a bounded operator, U ∈ C k a k-doped Clifford circuit; then the (C k , 4)fold channel for the k-doped Clifford circuit reads and P(2 N ) is the Pauli group on N qubits; T π are permutation operators corresponding to π ∈ S 4 , then Ξ k is the k-matrix power of the matrix Ξ, whose components read and the information about the operator O is all contained in the coefficients where W ± πσ are the generalized Weingarten functions for the Clifford group, introduced and discussed in App. A.1.
The proof of the theorem can be found in App. B.1.
For many purposes, it is important to know to what Φ C k (·) converges in the limit of infinite layers. Without substantial loss of generality, we consider the case of the non Clifford resources given by phase gates P θ with θ = π/2. We can thus establish the application: } is the single qubit computational basis, and for any θ = ±π/2 the (C k , 4)-fold channel equals the (U(d), 4)-fold channel in the limit k → ∞ is The proof can be found in App. B.2.
In the next sections, we apply these theorems to calculating the 8-point OTOCs and fluctuations of subsystem purity to find how these quantities approach the Haar-average on U(d) with k.

The 8-point OTOC
Consider four non-identity and non-overlapping Pauli operators A, B, C, D ∈ P(d). Then consider the unitary evolution of A U = U AU † in the Heisenberg picture and define an 8-point Out of Time Order Correlator (OTOC) as [3], defined in Eq.(3) We are interested in taking the twirling of the 8-point OTOC for a k-doped Clifford circuit, in order to find a necessary and sufficient condition for the exponential decay of the OTOC. Thanks to Theorem 1 we obtain Application 2 Let K ≡ T the single qubit T -gate, then the average of the 8-point OTOC over the k-doped Clifford circuit reads where Proof.-Starting from Eq.(15) we can write the 8-point OTOC for U as Taking the average over the k-doped Clifford U ∈ C k we have from the latter equation, the calculation is a straightforward, but tedious application of Theorem 1.
2 The following corollary explicitly shows the difference of the scaling of the 8-point OTOC for a pure Clifford circuit and for an universal circuit. As we shall see there is a marked difference in these scalings. As a direct consequence of Theorem 1 and Application 1, we obtain the following Corollary 1 Taking the average for U ∈ U(d) and C ∈ C(d) of the 8-point OTOC, one gets Proof.-The proof of Eq. (19) can be obtained from Eq.(16) in the limit k → ∞, in virtue of Theorem 1, while Eq.(20) can be obtained from Eq.(16) setting k = 0.
With the following statement we give the necessary and sufficient condition for the number of non Clifford gates needed to precisely simulate the behavior of the 8-point OTOC, and thus to simulate quantum chaos.
Proof.-Taking the difference in absolute value between Eq. (16) and (19) we get from here it's easy to see that one has the following condition which leads to the desired result. 2

Purity and its Fluctuations
In this section, we compute the fluctuations of a subsystem purity Eq.(4) for the output of the k-doped Clifford circuit U ∈ C k . To this end, we first apply Theorem 1 to calculate the average of the fourth tensor power of a pure state ψ, namely Φ sym is the projector onto the completely symmetric subspace of the permutation group S 4 and D sym = tr(Π (4) sym ). The coefficients a k , b k are given by The proof can be found in App. B.3. The evaluation of Eq.(25) becomes particularly simple if the gate K is a P θ -gate: Application 4 If the single qubit gate K is the P θ -gate, the coefficients c Q , c QQ ⊥ read: Then for any k we can write the coefficients a k , b k as See App. B.4 for the proof.

Corollary 3 For any
Proof.-The proof follows directly from Application 1; here we give an alternative version: setting f θ < 1 in Eq.(30) and taking the limit k → ∞ one gets Now, since the fourth tensor power of ψ U averages to -see Eq.(64) in App. A.3.1 for a proof - then by Application 3 the proof is complete. 2 In what follows, we calculate the purity and its fluctuations in a bipartite Hilbert space for the output state of a k-doped Clifford circuit, calculated above in Eq.(25). Consider then a bipartition of the N -qubit The averages over Unitary and Clifford group for the purity of the output ψ of a random quantum circuits are the same, namely This is a consequence of C(d) being a 3-design [15,16] (in fact, being a 2-design is sufficient), see App.
A.3.1 for a proof. Notice that the average purity does not depend on the input state. The fluctuations of the purity for the set x are defined as Since the fluctuations involve the fourth moment of the Haar measure, the fluctuations for U(d), C(d) are expected to be different. We have indeed, for This result is a consequence of Application 3 and Corollary 3. Notice that while the fluctuations of the purity for the unitary group again do not depend on the initial state, those for the Clifford group do. Notably, starting from completely factorized states, there is a marked difference whether the initial state ψ is a stabilizer state or a random product state.

Lemma 1
The fluctuations of the purity in the k-doped Clifford circuit, for The proof can be found in App. B.5.
Remark 1 For the undoped, k = 0, pure Clifford circuit, one finds Notice that, in the large d limit, and thus have the same order. However, the next corollary shows that -because of an exact cancellationthe fluctuations are very different in scaling with d.

Corollary 4 The fluctuations of the purity, for d
Proof.-Eq.(44) follows immediately from Lemma 1 by setting k = 0, while Eq.(45) can be found in [17]. 2 moreover, note that the rate of convergence is dictated by f θ , which reaches its minimum value for θ = π/4, that is the T -gate, cfr. Eq.(83). 2

Lemma 2
The fluctuations of the purity for a k-doped Clifford circuit, for d A = d B = √ d and ψ be a random product state read Proof.-The proof is straightforward and is left to the interested reader: by plugging (157) into Eqs. (26) and (25) and using Eq.(119) the calculation follows easily.

Corollary 6
The fluctuations of the purity for a non-doped Clifford circuit, for d A = d B = √ d and for ψ a random product state are Proof.-This result is obtained from Lemma 2 setting k = 0.

Conclusions and Outlook
In this paper, we showed that in a random Clifford circuit with N qubits, O(N ) non Clifford gates are both necessary and sufficient to simulate quantum chaos. As a consequence, quantum chaos cannot be efficiently simulated on a classical computer, as the cost for simulating such circuits is exponential in the non Clifford resources.
In perspective, there are several open questions. One could generalize many of these results by proving that an -approximate 2t-OTOC characterizes an -approximate t-design. Although the scaling is fixed to be O(N ), the actual number of non Clifford resources is undetermined and it would be of practical importance in obtaining approximate t-designs with a noisy, intermediate-scale quantum computer. One could thus study the optimal the arrangement of non Clifford resources. A related question is that of the onset of irreversibility in a closed quantum system in the sense of entanglement complexity [14] is driven by the doping of a Clifford circuit. Similarly, it would be interesting to show how the entanglement spectrum statistics converges with the doping [11].

A Haar averages: Unitary vs Clifford group
In this section we are going to display the explicit formula to average over the full unitary group and the full Clifford group without going into the group theoretic details. See [18,19] for the Haar integration and [20,21] for the Clifford integration formula.

A.1 Clifford group average
Starting from the result about the 4-th moment of the Haar average over the Clifford group in [20] we are going to prove an useful lemma.
where Q = 1 d 2 P ∈P(d) P ⊗4 and Q ⊥ = 1l ⊗4 − Q, while W ± πσ are the generalized Weingarten functions, defined as here λ labels the irreducible representations of the symmetric group S 4 , χ λ (πσ) are the characters of S 4 , d λ is the dimension of the irreducible representation λ, D + λ = tr(QP λ ) and D − λ = tr(Q ⊥ P λ ) where P λ are the projectors onto the irreducible representations of S 4 and finally T σ are permutation operators corresponding to the permutation σ ∈ S 4 .
Proof.-The projectors onto the irreducible representations of S 4 read Starting from the integration formula (32) in [20] we have At this point, we just define and the derivation is complete. 2 An important property that will be used throughout the paper is the following: another important property is that Q is a projector, namely Q 2 = Q. Another useful result, related to the generalized Weingarten functions is the proof comes from Eq. (53) and from Π (4) sym = (4!) −1 π∈S4 T π .

A.2 Unitary group average
Let O ∈ B(H ⊗t ) be a bounded operator on t-copies of H, then the Haar average reads [18,19] where T π is the permutation operator corresponding to the permutation π ∈ S t , the t-dimensional symmetric group and W πσ are the Weingarten functions defined as where D λ = tr(Π λ ).

A.3.1 The average purity
Let us calculate the average purity for the output state ψ U , for U ∈ U(d) or U ∈ C(d); indeed the result of the average for the two groups is the same because the Clifford group forms a unitary 3-design and being a t-design means being at-design for anyt ≤ t. Then, the average purity Since T σ ψ ⊗2 = ψ ⊗2 as long as ψ is a pure state, we have σ , see Sec. C.2 for a more rigorous treatment, we have Let ψ U be the output state of a quantum circuit U . Let us average the fourth tensor power of this output state for U ∈ U(d). Using formula (59) we have where we used the fact that T σ ψ ⊗4 = ψ ⊗4 for any permutation operator T σ and that πσ∈S4 W πσ T π = Π (4) sym /D sym , where D sym = tr(Π C k (O) be its output through the (C k , 4)-fold channel. Then (65) then since the averages over C i for i = 1, . . . , k are independent from each other, we can also write (66) The first Clifford average before inserting any single qubit K-gate reads where we have used Eq.(52). We can recast it as Now we need to apply the first K i1 -gate on the i-th qubit; noting that [T π , K ⊗4 i1 ] = 0 for all π ∈ S 4 , we have and then average over another Clifford layer, knowing that the Clifford operator only acts non trivially only on the operator where Ξ πσ and Λ πσ read we have defined the matrix Ξ omitting the subscript K i1 because, as shown in Lemma 6, it does not play any role. Thus, we have we can recast it as The latter relationship can be easily generalized to k layers as This concludes the proof. 2

B.2 Proof of Application 1
From theorem 1, the (C k , 4)-fold channel reads First of all let us write this equation in matrix form for the coefficients; define T a vector with components the permutation operators T σ , c the vector with components c π (O) and similarly for b, then Eq.(79) becomes Φ (4) where the · stands for the row by column product and (·, ·) for the usual scalar product between lists. Recall that for the Unitary group the (U(d), 4)-fold channel reads where W is the matrix with components the Unitary group Weingarten functions, cfr Eq.(60). In the following we prove that the first piece in Eq.(80) vanishes in the limit k → ∞, while the second returns the matrix W .
Lemma 4 For K = P θ ≡ |0 0| + e iθ |1 1|, the matrix Ξ, defined in Eq.(9) has the following properties • Ξ is symmetric; • Ξ has rank 6; • the eigenvalues read where µ(λ) stands for the algebraic multiplicity of the eigenvalue λ and The proof comes from direct calculation of the 24 × 24 matrix Ξ with K = P θ . 2 Since all the eigenvalues of Ξ are less than 1, lim k→∞ (Ξ k ) πσ = 0, for all π, σ (84) Defining the vector q having components tr(OQT σ ) and the vector t having components tr(OT σ ), from Eq. (12) and Eq.(13) we note that where W ± are the matrices with components the generalized Weingarten functions for the Clifford group, cfr. (53). Therefore taking the limit k → ∞ in Eq.(80) It is straightforward to check that because of Lemma 5 it is clear that q ∈ ker(ζ), which proves the theorem. 2

B.3 Proof of Application 3
We will make use of Theorem 1 Let us first compute the term πσ∈S4 c σ (ψ ⊗4 )(Ξ k ) πσ QT π ; note that [T π , Q] = 0. This is a fact that will be repeatedly exploited in this proof. First we prove that c σ = c independent from the specific permutation σ; from Eq.(12) we have where we used T π ψ ⊗4 = ψ ⊗4 for all π and π∈S4 W ± πσ = (4!D ± ) −1 . Then the sum can be written as let us prove that π∈S4 Ξ πτ does not depend on τ ; from Eq.(9) it is easy to see that where we have used π∈S4 W ± πσ = (4!D ± ) −1 and Π sym for any τ ∈ S 4 . The coefficients c Q and c QQ ⊥ are to be computed in a straightforward way; for the case K = P θ they are explicitly calculated in App. B.4. Since π∈S4 Ξ πτ does not depend on τ , the decomposition introduced in the last equality of Eq. (93) can be reiterated k times to obtain Now we compute the term πσ∈S4 Γ (k) πσ is defined in Theorem 1; since c π (ψ ⊗4 ) is independent from the permutation π we can write πσ∈S4 It is easy to see π∈S4 Λ πτ = c QQ ⊥ /D − does not depend on τ ; from this fact, we can use the same technique used above to compute τ ∈S4 sym (97) The last term we need to evaluate is π b π (ψ ⊗4 )T π ; as before, let us prove that b π (ψ ⊗4 ) does not depend Putting together Eqs.

B.4 Proof of Application 4
In this section we evaluate c Q = tr(K ⊗4 ij QK †⊗4 ij QΠ (4) sym ) for K = P θ ≡ |0 0| + e iθ |1 1|. As pointed out in Lemma 6, the position of the operator K ij does not affect the calculations, so in the following we analyze the case in which the operator K i acts on the first qubit i 1 . The term c Q can be rewritten as where R ∈ S 4N is a permutation operator whose action on a tensor product basis element is The adjoint action of the permutation R allows us to rewrite the operator Q as where X, Y, Z, I are single qubit Pauli matrices and for example Q X reads and Q d/2 = (d/2) −2 P ∈P(d/2) P ⊗4 . Similarly the adjoint action of R on K ⊗4 acting on the first qubit element of each copy of H it acts on the first four elements of RQR −1 . It is simple to see that the adjoint action of R on the symmetric projector Π (4) sym give us again the symmetric projector on the new permuted space and we denote it withΠ (4) sym . The term c Q can be rewritten as where we used the fact that [K, I] = [K, Z] = 0; then the first term of Eq.(105) reads We focus now on the second term of Eq.(105) where denotedT σ = RT σ R −1 and defined and In a similar fashion to what we have done in Sec. C.2 it is possible to see thatT σ = T (2) ; thus, the following equality holds It is easy to see that With all the previous consideration we are now ready to compute the coefficients 1 × tr(Q) = 16(cos 4 θ + sin 4 θ)tr(Q d/2 ) = 4(cos 4 θ + sin 4 θ)d 2 Therefore It is possible to calculate the extreme points of f θ . the maximum is f θ = 1 for θ = π/2, while the minimum is f θ ≈ 3 4 for θ = π/4.

C.2 Q decomposition and traces
It is interesting to prove an useful property of the operator Q, that we recall is defined as where σ ij ∈ P(2). It is possible to introduce a permutation S ∈ S 4N , whose action on a tensor product state is defined as The adjoint action of a permutation S on Q reads where Q 2 = σ∈P(2) σ ⊗4 . Then, let us show the adjoint action of S on a permutation operator between the 4-copies of H. Let T σ ∈ B(H ⊗4 ) a permutation operator between 4 copies of H ≡ C 2⊗N corresponding to σ ∈ S 4 ; written in terms of bras and kets it reads where we are denoting T (2) σ = i,j,k,l |σ(i)σ(j)σ(k)σ(l) ijkl| ∈ B(C 2 ) which is a permutation operator between 4 copies of a single qubit Hilbert space C 2 .
Lemma 6 Let K i and K j two identical single qubit gates with support on a different qubit, i and j respectively. Let T σ be a permutation operator between the 4-copies of H; the following equality holds Proof.-First of all Acting adjointly with the permutation operator S ∈ S 4N , defined in Eq.(141) on K ⊗4 i we have Then from the above equality and from Eq.(143) and Eq.(146) we have = tr(T (2) σ Q 2 KQ 2 K † )tr(T Proof.-First of all let us calculate tr(Qψ ⊗4 ) in the case ψ = |0 0| ⊗N . As proven in Sec. C.2 where S is a permutation operator defined in Eq.(141). Here Q 2 reads Now let us average tr(ψ ⊗4 Q) with the local-qubit Haar average. Let ψ ⊗4 loc be where supp(U i ) = C 2 for any i.