Efficient quantum algorithms for stabilizer entropies

Stabilizer entropies (SEs) are measures of nonstabilizerness or `magic' that quantify the degree to which a state is described by stabilizers. SEs are especially interesting due to their connections to scrambling, localization and property testing. However, applications have been limited so far as previously known measurement protocols for SEs scale exponentially with the number of qubits. Here, we efficiently measure SEs for integer R\'enyi index $n>1$ via Bell measurements. The SE of $N$-qubit quantum states can be measured with $O(n)$ copies and $O(nN)$ classical computational time, where for even $n$ we additionally require the complex conjugate of the state. We provide efficient bounds of various nonstabilizerness monotones which are intractable to compute beyond a few qubits. Using the IonQ quantum computer, we measure SEs of random Clifford circuits doped with non-Clifford gates and give bounds for the stabilizer fidelity, stabilizer extent and robustness of magic. We provide efficient algorithms to measure Clifford-averaged $4n$-point out-of-time-order correlators and multifractal flatness. With these measures we study the scrambling time of doped Clifford circuits and random Hamiltonian evolution depending on nonstabilizerness. Counter-intuitively, random Hamiltonian evolution becomes less scrambled at long times which we reveal with the multifractal flatness. Our results open up the exploration of nonstabilizerness with quantum computers.

Recently, SEs have also been related to various important properties of quantum systems.SEs probe errorcorrection [25] and measurement-induced phase transitions [26,27], as well as relate to the entanglement spectrum [28] and property testing [29,30].SEs are also connected to the participation entropy [31], which are helpful to understand Anderson [32] and many-body localization [33].Further, recent works established a fruitful connection between out-of-time-order correlators (OTOCs) and nonstabilizerness [30,34,35].OTOCs describe scrambling in quantum systems [36,37].However, OTOCs are challenging to measure directly and often require an inverse of the time evolution [38].Higher-order OTOCs and nonstabilizerness have been related to quantum chaos [34] and state certification [30].
The aforementioned properties make SEs highly interesting for experimental studies of quantum computers and simulators.However, the progress has so far been limited as all previously known measurement protocols for SEs scale exponentially with the number of qubits [14,39].
Here, we efficiently measure SEs with integer index n > 1 on quantum computers and simulators via Bell measurements on two copies of N -qubit quantum states.Our algorithms are practical to implement with O(n) copies and O(nN ) classical post-processing time, where even n requires also access to the complex conjugate of the state.We devise an efficient protocol to measure Clifford-averaged multifractal flatness and 4n-point OTOCs where for odd n we do not require an inverse time evolution.We study the interplay of nonstabilizerness and scrambling and show that the number of Clifford gates needed for OTOCs to converge depends on the number of T-gates.Further, we use the multifractal flatness to show that random Hamiltonian evolution stops being random for long evolution times.We also provide efficiently computable bounds to other nonstabilizerness monotones, which are otherwise intractable beyond a few qubits.Finally, we measure the Tsallis SE on the IonQ quantum computer and demonstrate SEs as efficient bounds for the robustness of magic, stabilizer extent and stabilizer fidelity.Our work introduces methods to uncover the key features that characterize the power of quantum computers and simulators.
SE.-For an N -qubit state |ψ⟩, the Rényi-n SE is given by [15] M where n is the index of the SE and P is the set of 4 N Pauli strings.The Pauli strings are N -qubit tensor products σ r = N j=1 σ r2j−1r2j with r ∈ {0, 1} 2N where σ 00 = I 1 , σ 01 = σ x , σ 10 = σ z and σ 11 = σ y with ℓ-qubit identity matrix I ℓ and Pauli matrices σ k , k ∈ {x, y, z}.M n is a faithful measure of nonstabilizerness for pure states, i.e.M n (|ψ STAB ⟩) = 0 only for pure stabilizer states |ψ STAB ⟩, and greater zero else [15].Further, SEs are invariant under Clifford unitaries U C with M n (U C |ψ⟩) = M n (|ψ⟩), where Clifford unitaries map any Pauli string σ to another Pauli string σ ′ via U C σU † C = σ ′ .Further, M n is additive with M n (|ψ⟩ ⊗ |ϕ⟩) = M n (|ψ⟩) + M n (|ϕ⟩).M n for n < 2 is not a monotone under channels that can map a pure state to another pure state, while the case n ≥ 2 remains an open problem.
Evaluating Eq. (1) requires an efficient measurement protocol for the n-th moment of the Pauli spectrum which on first glance appears challenging due to the summation over exponentially many Pauli strings. Algorithms.
-We now provide two algorithms to efficiently measure A n (|ψ⟩).First, we introduce Algorithm 1 which is efficient for odd n > 1.We write A n as the expectation value of an observable Γ ⊗N n acting on 2n copies of |ψ⟩ via the replica trick [16] A where Γ n = 1 2 3 k=0 (σ k ) ⊗2n .For even n > 1, 2 −N Γ ⊗N n is a projector with two possible eigenvalues ω ∈ {0, 2 N } as shown in the Supplemental Material (SM) A. In contrast, for odd n > 1 it is unitary and hermitian with eigenvalues ω ∈ {−1, 1}.This fact was previously pointed out in Ref. [40] for stabilizer testing.
To measure the operator Γ ⊗N n we transform the operator into a diagonal eigenbasis.We first recall the Bell transformation acting on two qubits U Bell = (H ⊗ I 1 )CNOT, where H = 1 √ 2 (σ x +σ z ) is the Hadamard gate, and CNOT = exp(i π Algorithm 1 utilizes this fact to provide an unbiased estimator for A n , where ⊕ denotes binary addition.While Eq. ( 4) involves 2n copies of |ψ⟩, the Bell transformation Eq. ( 4) can be written as tensor products.Thus, A n is evaluated using only Bell measurements on two copies of the quantum state, which requires only a 2Nqubit quantum computer.Then, via post-processing A n is computed as the parity of odd and even qubit index Bell measurement outcomes as derived in SM B. We apply Hoeffding's inequality to bound the number of copies as C = O(n∆ω n ϵ −2 ), where ϵ is the error and ∆ω n the range of eigenvalues of Γ ⊗N n .For odd n > 1, we have ∆ω n = 2 and C = O(nϵ −2 ).For even n, the eigenvalue spectrum of Γ ⊗N n diverges and we require an exponential number of measurements.In SM C we extend our algorithm to get gradients ∂ k A n via the shift-rule for variational quantum algorithms.Now, we provide Algorithm 2 which is efficient for any integer n > 1, however requires access to the complex conjugate |ψ * ⟩.We rewrite the SE as a sampling problem where Ξ(σ) = 2 −N ⟨ψ|σ|ψ⟩ 2 is the probability distribution of Pauli strings σ.The circuit for the algorithm is shown in Fig. 1 Tsallis SE.-We now define a measure of nonstabilizerness which we call the Tsallis-n SE [47] T They are a generalization of the linear SE T 2 [15] and the von- T n can be efficiently measured for integer n > 1 using our protocols.They are faithful measures of nonstabilizerness which are invariant under Clifford unitaries and related to Rényi SEs via Tsallis SEs lack the additive property of the Rényi SE, however our numerics suggest that Tsallis SEs may be a strong monotone which is a not necessary but highly desirable property for resource measures [48].Within extensive numerical optimization for N ≤ 6 qubits we were unable to find states that could violate strong monotonicity for the Tsallis-n SE for n ≥ 2 (see SM D).
Note that measuring the Rényi SE M n ∼ ln(A n ) with precision ϵ M requires O(n exp(M n )ϵ −2 M ) samples due to the logarithm (see SM D).Thus, M n is efficiently measurable as long as M n = O(log(N )).
Clifford-averaged OTOCs.-We now show how to efficiently measure 4n-point OTOCs of unitary U averaged over the Clifford group.The 4n-point OTOC for N -qubit Pauli strings σ and σ ′ is given by [30] otoc 4n (U, σ, σ ′ ) = 2 −N tr(σU σ ′ U † ) 2n (7) We find that otoc 4n averaged over the group of Clifford unitaries C N can be related to SE of U , which we define via the Choi state In particular, we have where the Pauli strings σ, σ ′ ∈ P/{I N } exclude the identity and the proof is found in SM K using results of Ref. [30].For odd n > 1, we can efficiently measure Eq. ( 8) via Algorithm 1.For even n > 1, we additionally require the complex conjugate |U * ⟩ for Algorithm 2. The complex conjugate of the Choi state can be efficiently prepared with access to U * or U † due to the ricochet property [46].Multifractal flatness.-Theparticipation entropy is given by I q (|ψ⟩) = k | ⟨k|ψ⟩ | 2q where |k⟩ are computational basis states, q > 0 and 0 ≤ I q ≤ 1 [32].The participation entropy quantifies the spread of the wavefunction over basis states, i.e.I q = 1 for computational basis states, while it is small when the state is delocalized over many computational basis states.The multifractal flatness F(|ψ⟩) = I 3 (|ψ⟩) − I 2 2 (|ψ⟩) measures the flatness of the distribution | ⟨k|ψ⟩ | 2 , i.e. we have F = 0 when the distribution | ⟨k|ψ⟩ | 2 is constant over its support, else we have F > 0. In particular, stabilizer states have F = 0 [49].
Recently, F averaged over C N has been proposed as F [31].This quantity describes the participation ratio averaged over all possible choices of basis states.F has been connected to SEs as follows [31] Thus, Algorithm 2 allows us to efficiently measure F(|ψ⟩) directly without the need of averaging over C N .Bounds on nonstabilizerness.-Wenow provide efficient bounds on three magic monotones, namely the robustness of magic R [8], stabilizer extent ξ [50] and the stabilizer fidelity [50] (see SM E).Computing R, ξ and F STAB requires solving an optimization program over the set of pure N -qubit stabilizer states.As the number of stabilizer states scales as O(2 N 2 ), these three measures in general become numerically infeasible beyond 5 qubits [8,51].
Our algorithms provide efficient bounds for integer n > 1 (see [17] or SM E): The bound can be tightened for n n [8,15].With methods from Ref. [40,52], we also prove a lower bound on F STAB for n > 1 (see SM F) The min-relative entropy of magic D min = − ln(F STAB ) can be seen as the distance to the nearest stabilizer state.We now argue that D min , Rényi SEs with n ≥ 2 and the recently introduced additive Bell magic B a [14] are closely related.In particular, we find evidence for respective upper and lower bounds independent of qubit number N (see SM G).Via numerical optimization we find 1.7M 2 ≳ D min ≥ 1 4 M 2 as well as 3.5M 2 ≳ B a ≳ 2.88M 2 for at least N ≤ 4, while similar bounds can also be found for larger n.Thus, D min , M n≥2 and B a can be seen as measures of nonstabilizerness that relate to the distance to the nearest stabilizer state.In contrast, the robustness of magic R and stabilizer extent ξ relate to the degree a state can be approximated by a combination of stabilizer states.They belong to a different class of nonstabilizerness measures as they cannot be upper bounded with D min or M n for n > 1/2 [17]. Demonstration.
-We now study SEs with Bell measurements on the IonQ quantum computer [14] using Algorithm 1 in Fig. 2. We investigate random Clifford circuits U C doped with N T non-Clifford gates where ) is the T-gate of the k-th layer acting on a randomly chosen qubit g(k).With increasing N T these states transition from efficiently simulable stabilizer states to intractable quantum states [34,53].To reduce noise, we compress the circuits into layered circuits composed of single-qubit operations and CNOT gates arranged in a nearest-neighbor chain configuration [14].The state prepared by the quantum computer is not pure but degraded by noise.However, SEs are faithful measures of nonstabilizerness only for pure states.Using measurements on the noisy state, we mitigate A n and T n from measurements on noisy states by assuming a global depolariziation error model (see SM H).
In Fig. 2a, we show T 3 with and without error mitigation for different N T , where for each value we average over 6 random instances of Eq. ( 12).We find that the results on the IonQ quantum computer with error mitigation closely match the simulated values.The Tsallis SE is zero for N T = 0, then increases with N T until it converges to the average value of Haar random states indicated as black dashed line.In Fig. 2b, we use the mitigated results for A 3 to compute upper and lower bounds for the stabilizer fidelity F STAB using Eq. ( 11).The measured result indeed gives valid bounds of the exactly simulated F STAB .We find that the upper bound is relatively tight, while the lower bound is non-trivial only for small N T .In SM I, we provide additional results for the IonQ quantum computer on measures of nonstabilizerness.While our error mitigation scheme assumes global depolarization noise, it works well on actual quantum computers which have more complicated noise profiles.In SM J, we simulate our error mitigation scheme for various unital and non-unital noise models, and find very good performance.As SEs are moments of exponentially many Pauli strings, self-averaging effects may explain the good performance.Scrambling.-We now study scrambling using the multifractal flatness F and OTOCs.In Fig. 3a, we show otoc 8 (U, σ x 1 , σ x 1 ) against d layers of Clifford gates doped with N T T-gates.We find that the OTOC decreases with d, converging to a minimum once the unitary is fully scrambled.This minimum is given by the Clifford averaged OTOC Eq. ( 8) and depends on N T .The d needed to converge depends on the number of T-gates, where for N T = 0 convergence is achieved for d ∼ 10, while higher N T requires larger d to converge.We observe similar convergence for F and other OTOCs in SM K.
In Fig. 3b, we study F for the evolution of a state |ψ(t)⟩ = exp(−iH GUE t) |0⟩ in time t using a random Hamiltonian H GUE drawn from the Gaussian Unitary ensemble (GUE).We observe that F initially increases, reaching a maximum at t ∼ 1.This is followed by a sudden dip to the Clifford-averaged multifractal flatness Eq. ( 9).This is hallmark of reaching deep thermalization or unitary design, where the system is indistinguishable from Haar-random dynamics [54].Counterintuitively, for longer (exponential) times F ramps up again, converging to a value above the Clifford-average.Here, the system stops being fully random due to dephasing of energy eigenvalues [54].In SM K, we show how to measure F and approximate GUE Hamiltonians using a Hamiltonian of random Pauli strings which can be implemented in experiment.
Discussions.-Weshow how to efficiently measure SEs with a cost independent of qubit number N , which is an exponential improvement over previous protocols [14,39].For integer n > 1, our protocol is asymptotically optimal with the number of copies scaling as O(nϵ −2 ) and the classical post-processing time as O(nN ϵ −2 ) with additive error ϵ.The protocol is easy to implement using Bell measurements which have been demonstrated for quantum computers and simulators [55][56][57].We note that our approach is distinct from the previously introduced Bell magic [14] as shown in SM L.
Our measurement protocol allows for efficient experimental characterization of different important properties of quantum states.We demonstrate an efficient bound on nonstabilizerness monotones which otherwise are hard to compute beyond a few qubits.These monotones serve as lower bounds on state preparation complexity and characterize the runtime of classical simulation algorithms [8,50].Further, we show how to efficiently measure Clifford-averaged 4n-point OTOCs.Our protocol has the advantage that it does not require implementing time-reversal for odd n > 1 which can be a challenge [36].Our protocol can measure higher order OTOCs which promise to reveal more features compared to the usually considered 4-point OTOCs [58,59].Our methods allow direct experimental study phase transitions in SE which have been found for purity testing [29] and quantum error correction [25].Finally, we enable certification of magic gates in fault-tolerant quantum computers, where the SE could be directly evaluated from recent experimental data [60].
We use our methods to study scrambling in Clifford circuits doped with T-gates and random Hamiltonian evolution.OTOCs not only measure scrambling, but also depend on nonstabilizerness in a non-trivial way [61].We can disentangle these two effects by measuring the Clifford-averaged OTOC.We study when Clifford circuits doped with T-gates become fully scrambled, revealing that the depth depends on the number of T-gates.We also study the scrambling with random Hamiltonians.Notably, random Hamiltonian evolution deep thermalizes at intermediate times, becoming indistinguishable from Haar random unitaries [54] which we observe via the convergence of the multifractal flatness to its Clifford-averaged value.Counter-intuitively, the evolution becomes less random again for long times due to the dephasing of its energy eigenstates [54].While non-local OTOCs and SEs lack clear signatures of this effect (see SE L), we find that multifractal flatness and same-site OTOCs exhibit clear gaps to their Clifford-average.
Future work could find efficient protocols for even n without the need of complex conjugation and tighten the lower bound of SEs for the stabilizer fidelity.
The code for this work is available on Github [62].
Note added: Before acceptance of this letter, the monotonicity of Rényi SE and strong monotonicity of Tsallis SE has been proven for n ≥ 2 [63].

Supplemental Materials
We provide additional details on the efficient measurement of stabilizer entropies and prove bounds on stabilizer fidelity.Further, we show how to compute gradients, mitigate errors and provide additional results for the IonQ quantum computer.Finally, we study nonstabilizerness and scrambling using Clifford-averaged multifractal flatness and OTOCs, as well as provide an efficient measurement scheme for multifractal flatness.We now discuss the eigenvalue spectrum of the observable that measures the SE.The SE is defined as a sum over powers of expectation value of the Pauli operator.By assuming access to n copies, we can write it in terms of an SE observable We now derive the parity check rules to evaluate A n using Algorithm 1.To evaluate A n , we note it is composed as a tensor product of N operators which we evaluate one by one with a simple rule.In particular, for the operator U ⊗n Bell Γ n U ⊗n

Contents
Bell † and the corresponding state |ψ⟩ ⊗2n , we reorder the position of the qubits (r 1 , r 2 , . . ., r 2n ) → (r 1 , r 3 , . . ., r 2n−1 , r 2 , r 4 , . . ., r 2n ) such that the odd-indexed qubits occupy the first half, and the even-indexed qubits the second half of the register.Then, we find for the transformed operator for odd n the simple form and for even n Here, the expectation value ⟨x⟩ = ⟨x| (1 − σ z ⊗n ) |x⟩ for n-dimensional computational basis state |x⟩ is ⟨x⟩ = 2 when x has odd parity, and 0 otherwise.In contrast, ⟨x| (1 + σ z ⊗n ) |x⟩ is 2 when x has even parity, and is 0 otherwise.Thus, we see that U ⊗n Bell Γ n U ⊗n Bell † can be evaluated by checking the parity of the odd-indexed and even-indexed qubits.
We bound the maximal number L of measurement steps needed to measure SE with Hoeffding's inequality.The failure probability ∆ to get an error ϵ between expectation value A n and estimation Ân is given by where ∆ω n is the range of eigenvalues of Γ ⊗N n .To estimate A n within ϵ accuracy and δ failure probability we require at most measurement steps.Each step of the algorithm uses 2n copies of the state as seen in Algorithm 1, thus the total number of copies is C = 2Ln.For odd n > 1, we have ∆ω n = 2 and the number of copies of |ψ⟩ scales as C = O(nϵ −2 ).For even n, the eigenvalue spectrum of Γ ⊗N n diverges and Algorithm 1 of the main text requires in general an exponential number of measurements.

Supplemental Materials C: Parameter-shift rule for gradient of SE
We now derive the parameter-shift rule for observables that act on multiple copies of a state.First, we regard the generator G of the unitary operator G(µ) = e −iµG has two distinct eigenvalues, we can shift the eigenvalues to ±r.Note that any single qubit gate is of this form, which implies G 2 = r 2 I, where I is the identity matrix.The Taylor series of G(µ) shows Thus the following identity holds 2 σn given by some Pauli strings σ n .For any operators U , O, and V we have where h.c. is the hermitian conjugate of the preceding terms [64].
To measure SE, we calculate the expectation value over K copies of state |ψ⟩ ∈ C 2 N with in respect to an operator O (K) ∈ C 2 N K .For K = 1, i.e. measurements on a single quantum state, the shift rule is given by , where e k is the kth unit vector [64].
We now derive the shift-rule for general K.The derivative of the quantum state |ψ⟩ is given by where From the product rule, the K-copy derivative of ⟨O⟩ is given by In the above, the states are freely rearranged while the factor of K emerges from the product rule.We now define Then, applying Eq. (S6) in the second row gives For any Pauli strings σ k , we can use Eq.(S5) to find where we have introduced the kth unit vector e k and absorbed the exponential into the definition of the kth parameterized rotation: We can now get the n-th moment of the Pauli spectrum simply by setting K = 2n and performing Bell measurements.
To compute the gradient of the Tsallis SE in respect to the kth parameter, we have where For the Rényi SE, we have Now, we present the algorithm to efficiently compute an unbiased estimator of the gradient of the SE on quantum computers for odd n > 1.The algorithm is detailed in Algorithm 3. The algorithm computes the gradient of A n using the shift-rule.To calculate the gradient, one performs Bell measurements on a parameterized quantum state shifted by ±π/2 with |ψ(θ ± π/2)⟩ |ψ(θ)⟩, as well as n − 1 Bell measurements on states without shift |ψ(θ)⟩ |ψ(θ)⟩.Then, use the same procedure as Algorithm 1 in the main text to estimate K ± , where its difference scaled by n gives us ∂ k A n .

Supplemental Materials D: Strong monotonicity and measuring Rényi SEs
A not necessary, but useful property of nonstabilizerness is strong monotonicity where the measure is on average non-increasing under computational-basis measurements on a set of k qubits [48].This property demands that T n (|ψ⟩) ≥ λ p λ T n [|ψ λ ⟩] when using the projector Π λ = |λ⟩⟨λ| ⊗ 1 1 N \k onto the computational basis state λ with corresponding probability p λ = ⟨ψ| Π λ |ψ⟩ and post-measurement state |ψ λ ⟩.The Rényi SE M n is not a strong monotone for all n [17].For n < 2, we can show that the Tsallis SE T n is not a strong monotone using the counter-example from Ref. [17].However, for n ≥ 2 we are unable to find any example where strong monotonicity is violated.Using extensive numerical optimization for the strong monotonicity condition, we find numerical evidence that T n for n ≥ 2 is a strong monotone for at least N ≤ 6 qubits.We use numerical optimization of a general parameterized N -qubit state |ψ(α, β)⟩ = k α k + iβ k |k⟩.Here, we minimize the condition of strong monotonicity where strong monotonicity is violated when ∆T n < 0. Numerically, we do not find states that violate strong monotonicity for the Tsallis-n SE for n ≥ 2, while we easily find counter-examples for the Rényi-n SE.
We now discuss the scaling of measuring the Rényi SE.We can estimate A n = 2 −N σ∈P ⟨ψ| σ |ψ⟩ 2n with additive precision using O(nϵ −2 ) samples with our algorithms.The Rényi SE involves the logarithm of A n , i.e.M n ∼ ln(A n ).Given A n estimated with additive precision ϵ, we now consider the cost of estimating M n with additive precision ϵ M .Given exact Ān and Mn , we assume from finite samples we have error ϵ in estimating A n , which gives us We now assume ϵ ≪ Ān and get The moment of the Pauli spectrum A n for n > 1 upper bounds the stabilizer fidelity, which was previously proven in Ref. [17] via the Rényi SE.Here, we reproduce the proof for completeness and adapt it to our efficient measurement algorithms.First, we note that for any state |ψ⟩ with stabilizer fidelity F STAB (|ψ⟩) one can find a Clifford unitary U C with where One can see this immediately from In Eq. (S4), we denote {|k⟩} the the computational basis states.Next, we have Here P is the set of all Pauli strings while P z is the set of Pauli strings which contains strings with 1 1 and σ z only.
We have used the convexity inequality We insert the definition of T n to finally get which concludes the proof of the upper bound.Furthermore, we note the well known relationship between R, ξ and which combined gives us the final bound: Supplemental Materials F: Proof of lower bound of stabilizer fidelity Here, we prove that the nth moment of the Pauli spectrum A n is a lower bound of the stabilizer fidelity F STAB .
The main ingredient is the relationship proven in [40,52] where Q = {σ ∈ P : ⟨ψ| σ |ψ⟩ 2 > 1 2 }.We now have In the first line we used the definition of Q, and in the second line we used Markov's inequality which holds for n > 1.

Supplemental Materials G: Relationship of measures of nonstabilizerness
Here we study the relationship of different measures of nonstabilizerness.In particular, we study the Rényi SE M n , the min-relative entropy of magic D min = − ln(F STAB ), the log-free robustness of magic LR = ln(R) and the maxrelative entropy of magic ln(ξ).We also consider the additive Bell magic [14] which is defined and further discussed in SM L.
We now study the N -qubit product state We study the scaling of this state with s.We show the scaling in Fig. S1.For D min , B a and M n with n ≥ 2 we find that these measures of nonstabilizerness scale as ∝ s 2 N .Defining θ = s √ N , we find for all these measures ∝ θ 2 .This indicates that these measure are closely related.We study the relationship between these measures further by finding respective upper and lower bounds.We numerical maximize and minimize the ratios D min /M n and B a /M n over all pure states for a given qubit number N .This gives us a lower and upper bound between these measures.Due to numerical complexity of computing D min and B a exactly, we can optimize up to N ≤ 4. We find that the bounds barely change with N , indicating that they are likely valid even for higher qubit numbers.In particular, we find 1.7M 2 ≳ D min ≥ 1 4 M 2 as well as 3.5M 2 ≳ B a ≳ 2.88M 2 for at least N ≤ 4. In contrast, for LR, D max and M 1/2 we find that these measures scale as ∝ sN and thus ∝ θ/ √ N .These measures show a different scaling with N compared to the measures that relate to geometric distance such as D min .This shows that LR, D max and M 1/2 do not support upper bounds in respect to M n for n > 1/2.As there are no known efficient measurement protocols for M 1/2 , it is likely that one cannot find efficiently measurable upper bounds for D max and LR.

Supplemental Materials H: Error mitigation
Noise in the quantum computer distorts the true value of the SE.We now propose how to mitigate noise without requiring additional measurements.Here, we assume that a pure state |ψ⟩ is subject to a global depolarizing channel with probability p, resulting in the noisy state ρ dp = (1 − p)|ψ⟩⟨ψ| + pI N 2 −N .The depolarising probability p can be deduced by the measurement of purity tr(ρ 2 dp ) as shown in Ref. [14] p = 1 − Note that tr(ρ 2 dp ) can be efficiently measured with Bell measurements [65].In particular, the purity can be measured via tr(ρ 2 dp ) = A 1 using Algorithm 1 in an efficient manner.SEs for depolarized state ρ dp can be expressed using the expectation value of operator Γ ⊗N n = 2 −N σ∈P σ ⊗2n acting on 2n copies of the mixed state ρ dp where A mtg n is the mitigated expectation value using the measurement result affected by depolarizing noise A dp n .We find We then substitute the above expression to calculate Similarly, we have for the Rényi SE

Supplemental Materials I: Further IonQ quantum computer results
We present here further experimental results on nonstabilizerness.In Fig. S2, we demonstrate the hierarchy of bounds between the measures of nonstabilizerness, where A 3 is measured in experiment.
In Fig. S3 we demonstrate our lower bounds for robustness of magic R and stabilizer extent ξ using our Algorithm 1 for the Pauli spectrum moments A n = 2 −N σ∈P ⟨ψ| σ |ψ⟩ 2n .We compute the bounds using the error mitigated measured A 2 and A 3 using Algorithm 1 of the main text.We show the upper bound n and the lower bounds which is the improved bound only valid for R.For A 2 , Algorithm 1 is not efficient, however for our chosen qubit number and measurement samples we are able to achieve sufficient accuracy.
We find that the exactly simulated bounds closely match the result from the IonQ quantum computer.We find the bounds for A 2 and A 3 show similar result, except for R where we find that A 2 gives a better bound.
Finally, we also demonstrate measurement of the Tsallis entropy T 5 in Fig. S4.We find similar behavior to T 3 as measured in the main text.

Supplemental Materials J: Error mitigation for different noise models
We now study the performance of our error mitigation strategy for different noise models.Our error mitigation strategy assumes global depolarization noise, which is a simple noise model.Our experiment indicate that it works well even for the noise of the IonQ quantum computer which is known to have a more complex noise model.Here, we provide further simulations with various noise models, finding that our model performs surprisingly well, both for unital and non-unital noise models.
We simulate random Clifford circuits constructed from layers of single-qubit rotations and CNOT gates arranged in a nearest-neighbor configuration, where we can dope the circuit with N T T-gates.After every gate, we apply a noise channel with noise strength p on the qubits the gate acted on.
We study the following three noise models, which have the following channel description for each qubit: For unital In Fig. S5c,d, we show amplitude dampening noise for different number of qubits N .We find that the reduction in error due to error mitigation is independent of qubit number for Clifford circuits, while for doped Clifford circuits the error decreases even further with N .This indicates that our error mitigation strategy can work well even for more qubits.

Supplemental Materials K: Nonstabilizerness, scrambling, multifractal flatness and OTOCs
Here, we study the relationship of SEs, scrambling, multifractal flatness and OTOCs for various circuit and Hamiltonian evolution models.
First, we review the properties as introduced in the main text.First, we have the n-Rényi SE of state |ψ⟩ as with the n-th moment of the Pauli spectrum where P is the set of all Pauli strings with sign +1.We can also define the n-Rényi SE M n (|U ⟩) for unitary U via its unique Choi state |U ⟩ = I N ⊗ U |Φ⟩ with the maximally entangled state |Φ⟩.
First, we note that any Clifford where from second to third line we used the fact that random Clifford transformations of the form U C σ a U † C map any Pauli string σ a ∈ P/I N to other Pauli strings (excluding identity I N ) with the same frequency, as well as |P/I N | = 4 N − 1.In the third line we used otoc 4n (U, σ, I N ) = otoc 4n (U, I N , σ) = δ σ,I N , and in the final line we used Eq.(S7).
We now study the 2-Rényi SE M 2 , the 8-point OTOC otoc 8 (U, σ, σ ′ ) and the multifractal flatness F for various models.For F, we study the flatness in regard to the state U |0⟩.

Random circuits
First, we study random circuits composed of d layers of random single-qubit rotations and nearest-neighbor CNOT gates arranged in a nearest-neighbor chain.These circuits are known to approximate Haar random unitaries in linear circuit depth [66].
We show M 2 , F and otoc 8 against d in Fig. S6.We show M 2 against d in Fig. S6a, observing a rapid increase with d which converges approximately at d ∝ N depth.We observe that the multifractal flatness in Fig. S6b converges to its Clifford-averaged value with d.Similarly, we find in Fig. S6c,d that same-site OTOC otoc 8 (U, σ x 1 , σ x 1 ) and non-local otoc 8 (U, σ x 1 , σ z N ) converge to its Clifford-average values.Curiously, we find that OTOCs require higher d to converge than multifractal flatness or SE, indicating that they are more sensitive to scrambling.

Random clifford + T
Next, we study random Clifford circuits drawn from the Clifford group C N doped with N T T-gates which are given as We show M 2 , F and otoc 8 against N T in Fig. S7.The dashed line is F and otoc 8 averaged over random Clifford unitaries computed using the SE M 2 .We find that the observed values match the values predicted via Eq.(S7) and Eq.(S4).Note that all otoc 8 for any pair σ, σ ′ ∈ P/I N yield the same value due to the scrambling with U C .

Fixed depth clifford + T
Next, we study in Fig. S8 fixed depth Clifford circuits doped with T-gates.In particular, we simulate a d layer circuit, where each layer is composed of random single-qubit Clifford gates on each qubit and CNOT gates arranged in a nearest-neighbor chain.This layered structure is repeated d times.Now, we randomly insert N T T-gates in this circuit at random positions.In this circuit, the degree of scrambling depends on the number of Clifford layers d, while the nonstabilizerness depends on N T .In Fig. S8a, we find that M 2 converges very fast with d, where the final value depends only on N T .In Fig. S8b, we study flatness F. We find convergence to the minimal value given by the Clifford averaged value already for d ∼ 5 layers.We observe that for small N T and intermediate d, F is smaller than measure of scrambling.
Our fixed depth Clifford + T circuits can be implemented directly in experiment.This can be realized with current noisy quantum computers [67], and for a low number of qubits even with recently developed fault-tolerant quantum computers [60].

Evolution with GUE Hamiltonian
Next, in Fig. S9 we study the evolution with Hamiltonians H GUE sampled randomly from the Gaussian Unitary ensemble (GUE).We normalize the Hamiltonian with a prefactor 2 −N/2 to restrict the eigenvalues of the Hamiltonian between [−2, 2] to get dynamics that is independent of N [54].We observe that SE in Fig. S9a increases until converging to its maximum around t ∼ √ N .However, the dynamics of F and otoc 8 shows more structure.In Fig. S9b, an initial increase with t finds a characteristic peak around t ∼ 1.This is followed by a sudden dip to a low value.This dip value is given by the Clifford-averaged multifractal flatness.After longer times F increases again, converging to a larger value above the dip value.We observe similar dip behavior also for the otoc 8 (U, σ x 1 , σ x 1 ) in Fig. S9c, reaching a minimal value which is given by the Clifford-averaged OTOC.At larger times the OTOC ramps again and converges to a larger value.Curiously, the duration of the dip is shorter for the OTOC compared to F, again indicating that OTOCs are more sensitive to the degree of scrambling.In Fig. S9d we regard otoc 8 (U, σ x 1 , σ z N ) the OTOC of observables at different points in space.Here, we find that the dip-ramp behavior is not clearly visible.
For GUE evolution, the dip behavior has been first noted in Ref. [54] for the frame potential and 2 and 4 point OTOCs.In Ref. [54] it has been shown that at the dip the ensemble of GUE evolved Hamiltonians matches a kdesign.Here, k-design refers that the ensemble of evolution unitaries is indistinguishable up to the kth moment to Haar random unitaries.However, at longer times it is not a k-design anymore.In particular, there is a characteristic which decay over long times t.We find that for short times, the initial increase, peak and following decrease of F are universal features of both Heisenberg evolution and GUE evolution.However, the dip assumes in general not the Clifford-averaged multifractal flatness, and converges to much larger F at long times.In Fig. S11c,d we study OTOCs as function of t.We note the pronounced oscillations, which are absent in the GUE.While the Heisenberg model is ergodic for small W , it still shows widely different behavior from the GUE evolution, and does not converge to its Clifford-average values.
general the protocols to measure OTOC require deeper circuits than measuring the multifractal flatness due to the need of forward and backward evolution.

Supplemental Materials L: Bell measurements, stabilizer entropy and Bell magic
Bell measurements are a powerful measurement scheme to estimate non-linear properties of quantum states.One takes two copies |ψ⟩ |ψ⟩ and transforms them into the Bell basis via U Bell as described in the main text.Sampling in the Bell basis gives an 2N bit outcome r with a probability [41] where σ r is a Pauli string.Bell measurements can be used to measure the purity, fidelity (as they realize a destructive SWAP test [65]), entanglement [65], coherence, imaginarity [45] and estimate the absolute values of any Pauli expectation value as a type of shadow tomography [56].Further, Bell measurements can be used to learn stabilizer states [41].All these properties can be extracted at the same time by using different post-processing steps.
For example to measure purity tr(ρ 2 ), one performs a logical AND on outcome r between each qubit pair corresponding to the first and second copy, then compute its parity [65].
For learning stabilizers, the core routine is Bell difference sampling [40].Here, one performs two Bell measurements with outcomes r, r ′ and takes their difference q = r − r ′ .
The probability of outcome q for Bell difference sampling is given by [40] Q(q) = This commuting property is used in Bell magic B to generate a measure of nonstabilizerness [14] where the infinity norm is zero ∥[σ r , σ q ]∥ ∞ = 0 when the two Pauli strings commute [σ r , σ q ] = 0, and ∥[σ r , σ q ]∥ ∞ = 2 otherwise.Correspondingly, the additive Bell magic is defined as B a = − log 2 (1 − B).Due to the commuting property, only for pure stabilizer states |ψ C ⟩ we have B(|ψ C ⟩) = 0, else it is greater zero.One can also check that it is invariant under Clifford unitaries.We also note that the complexity of the error mitigation scheme for Bell magic scales as 1/(1 − p) 8 where p is the depolarization probability.In contrast, for stabilizer entropy it scales as 1/(1 − p) 2n , e.g. for n = 3 we have 1/(1 − p) 6 .The measurement protocols for stabilizer entropy uses a different method to measure nonstabilizerness.In particular, one has to measure the operator Γ ⊗N = σ∈P σ ⊗2n which measures the n-th moment of the Pauli spectrum.This operator is diagonalized by Bell transformations as shown in SM B. By appropriately post-processing of Algorithm 1 in the main text one can then measure Our Algorithm 2 uses yet another approach to measure the stabilizer entropy.It samples directly from the Pauli spectrum (via the complex conjugate |ψ * ⟩ in a Monte-Carlo fashion and then averages over the sampled Pauli strings via

2 . 4 Sample r ∼ | ⟨r|η⟩ | 2 5 b = 1 6 for 7 Prepare 8 b = b • λ 9 end 10 An=Figure 2 .
Figure 2. Measurement of nonstabilizerness for quantum states generated by Eq. (12) with random Clifford circuits doped with NT T-gates on the IonQ quantum computer.a) We show Tsallis SE T3 with and without error mitigation as well as exact simulation.Dashed line is average value for Haar random states.Dots represent mean value and error bars the standard deviation taken over 6 random instances of the circuit.We have N = 3 qubits, 10 3 Bell measurements and a measured depolarization error of p ≈ 0.1.b) We show upper and lower bounds on FSTAB via Eq.(11) evaluated using the error mitigated T3 as blue and green dots as well as simulation of the bound as dashed lines.The orange dots show simulations of FSTAB.

Figure S1 .
Figure S1.Scaling of different measures of nonstabilizerness for state Eq.(S1) as function of s.

Figure S2 .Figure S3 .
Figure S2.We demonstrate the hierarchy of bounds Eq. (S13) for different measures of nonstabilizerness using error mitigated A3 from IonQ experiment as well as simulations of F −1 STAB , stabilizer extent ξ and robustness of magic R. Quantum states generated by random Clifford circuits doped with NT T-gates.

Figure S4 .
Figure S4.Measurement of T5 with and without error mitigation.Dashed line is average value of T5 for Haar random states.
unitary U C applied to a Pauli string σ ∈ P/I N (except identity I N ) yields another Pauli string σ ′ = U C σU † C ∈ P/I N .This mapping is bijective, i.e. each Pauli is mapped to exactly one other Pauli string.Further, the identity is mapped back onto itself, i.e.U C I N U † C = I N .Now, we define the OTOC averaged over random Clifford unitaries E UC,U ′ C ∈C N [otoc 4n (U C U U ′ C , σ a , σ b )] where we have any σ a , σ b ∈ P/I N .We find

32 d
Figure S8.Circuit with d layers of Clifford gates with NT T-gates injected.We study 2-Rényi SE M2(|U ⟩), the 8-point OTOC otoc8(U, σ x 1 , σ x 1 ) and the multifractal flatness F(U |0⟩).We show data as function of number of number of Clifford gate layers d, with varying number of T-gates NT. a) M2 against d.b) F against d.Dashed line is relationship between SE and Clifford averaged multifractal flatness Eq. (S4) for large d.c) otoc8(U, σ x 1 , σ x 1 ) against d, where dashed line is the Clifford-averaged OTOC given by Eq. (S7) for large d.d) otoc8(U, σ x 1 , σ z N ) against d.All results are for N = 4 qubits averaged over 30000 random instances.

A n = 2
that the fact that Pauli of stabilizer states commute is not used at all, in contrast to Bell magic.Instead, stabilizer entropy is a measure of the moment of the Pauli spectrum determine nonstabilizerness.In particular, stabilizer states have a peaked Pauli distribution with either | ⟨ψ| σ |ψ⟩ | 2 = 0 or | ⟨ψ| σ |ψ⟩ | 2 = 1.In contrast, highly magical states have a broad distribution | ⟨ψ| σ |ψ⟩ | 2 ≈ 4 −N .