Deterministic Fast Scrambling with Neutral Atom Arrays
Tomohiro Hashizume, Gregory Bentsen, Sebastian Weber, Andrew J. Daley
DDeterministic Fast Scrambling with Neutral Atom Arrays
Tomohiro Hashizume, ∗ Gregory S. Bentsen, ∗ Sebastian Weber, and Andrew J. Daley Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, UK Martin A. Fisher School of Physics, Brandeis University, Waltham MA, USA Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology,University of Stuttgart, 70550 Stuttgart, Germany (Dated: March 1, 2021)Fast scramblers are dynamical quantum systems that produce many-body entanglement on atimescale that grows logarithmically with the system size N . We propose and investigate a familyof deterministic, fast scrambling quantum circuits realizable in near-term experiments with arraysof neutral atoms. We show that three experimental tools – nearest-neighbor Rydberg interactions,global single-qubit rotations, and shuffling operations facilitated by an auxiliary tweezer array – aresufficient to generate nonlocal interaction graphs capable of scrambling quantum information usingonly O (log N ) parallel applications of nearest-neighbor gates. These tools enable direct experimen-tal access to fast scrambling dynamics in a highly controlled and programmable way, and can beharnessed to produce highly entangled states with varied applications. Quantum information scrambling describes a processin which initially localized quantum information is de-localized by the dynamics of a many-body system andencoded into a many-body entangled state [1–4], therebyeffectively hiding the information from local observers.Fast scramblers saturate the conjectured lower bound t ∗ ∼ log N on the timescale necessary to scramble quan-tum information, scaling logarithmically with the sys-tem size N . Fast scrambling dynamics can rapidly gen-erate Page-scrambled states, pure quantum states of amany-body system whose reduced density matrix ρ A ismaximally mixed for almost all subsystems A of size | A | < N/ fast scrambling quantum channels [2–4]. Using suchshuffling techniques for long-lived ground-state atomicqubits allows for the generation of highly non-local in-teraction graphs with only global rotations and nearest-neighbor Rydberg interactions. Though shuffling op-erations will generally be slower than Rydberg gates,limiting the number of gates to O [log( N )] minimisesthe primary sources of noise and decoherence, whicharise from laser excitations to Rydberg levels [20, 25–29].The simplest versions of these circuits efficiently producemany-body-entangled graph states [30], known computa-tional resources for measurement-based quantum compu-tation [31, 32], quantum metrology [33], quantum error-correction [34], and quantum cryptography [35]. More FIG. 1.
Fast scrambling via quasi-1D shuffling. (a)Neutral atoms (red dots, blue circles) trapped in a static 1Doptical lattice (gray boxes) can be rapidly rearranged via atwo-step shuffling operation R (i-iii) facilitated by an auxil-iary 1D tweezer array (bottom-left). Iterated shuffling andnearest-neighbor Rydberg interactions yield effective interac-tions on highly nonlocal coupling graphs such as the m -regularhypercube graph Q m (b). More generally, circuits (c) com-posed of shuffles (blue), nearest-neighbor controlled- Z opera-tions (red), and global rotations (purple) can be harnessed togenerate Page-scrambled quantum states in m iterations, orstrongly-scrambling quantum channels in 2 m iterations. sophisticated circuits built with the same experimentaltools yield strongly scrambling quantum channels capa-ble of robustly protecting quantum information againstmulti-qubit erasure [9, 36–38].Below we analyze iterated (Floquet) circuits built withthese tools both for the idealised unitary case and the dis-sipative case expected in realistic implementations. Wedemonstrate that initially-separable states can be Page-scrambled using only m ≡ (cid:100) log N (cid:101) nearest-neighbor in-teraction layers and construct deterministic circuits withonly 2 m interaction layers that strongly scramble quan-tum information regardless of the input state. a r X i v : . [ qu a n t - ph ] F e b The basis for our protocol is the possibility to realize afamily of sparse nonlocal coupling graphs via a quasi-1Dshuffling procedure (Fig. 1a) on atoms in optical latticesfacilitated by an auxiliary programmable 1D tweezer ar-ray. Straightforward stretching and interleaving tweezeroperations [39–42] (Fig. 1a(i-iii)) can be used to rapidlyshuffle the atomic positions. For N = 8 these motionsexecute the permutation R = (cid:18) (cid:19) (1)with atoms labelled by i = 0 , , . . . , N −
1. More gen-erally, for system sizes N = 2 m with m an integer, a perfect shuffle or Faro shuffle operation [43, 44] executesthe nonlocal mapping i (cid:48) = R ( i = b m . . . b b ) = b b m . . . b , (2)which cyclically permutes the bit order of the atomicindex i = b m . . . b b written in binary such that theleast significant bit b of i becomes the most signif-icant bit of R ( i ). The shuffling operation R , alongwith its inverse R − and generalizations thereof [45],are built on established tweezer-assisted techniques fordefect-removal in atom arrays [40–42], and can be imple-mented rapidly using a pair of Acousto-Optic Deflectors(AOD) in crossed configuration and driven by indepen-dent RF signals f x , f z (Fig. 1a, bottom-left).Repeated shuffling operations R dramatically rear-range the atomic positions. As a result, the propagationof quantum information is no longer constrained by theunderlying 1D geometry of the fixed optical lattice. Thesimplest iterated circuit E Q m ≡ [ R · CZ (even) ] m gener-ates effective controlled- Z interactions on the m -regularhypercube graph Q m [46, 47], a highly nonlocal, sparsely-connected coupling graph shown in Fig. 1b. These nonlo-cal couplings allow many-body entanglement to be builtup rapidly and efficiently using far fewer Rydberg inter-action layers than would be needed in strictly 1D systemswithout shuffling. For example, given N = 2 m qubits ini-tialized in the product state (cid:81) i | + (cid:105) i = (cid:81) i ( | (cid:105) + | (cid:105) ) i / √ E Q m produces the Page-scrambled graph state | Q m (cid:105) after only m interaction layers CZ (even) [33, 45].More sophisticated circuits built using the same exper-imental tools (Fig. 1c) can robustly scramble quantuminformation irrespective of the input state. By includ-ing global Hadamard H and Phase P rotations, one canimplement a strongly-scrambling circuit E s ≡ [ R − · CZ (odd) · H · P ] m [ R − · CZ (even) · H · P ] m (3)that yields widespread many-body entanglement afteronly 2 m interaction layers CZ (even) , CZ (odd) for arbitraryinput states, as demonstrated by numerical studies ofClifford circuits (Fig. 2)[45, 48, 49]. For N = 128 ini-tially z -polarized qubits, randomly-chosen subsystems A FIG. 2.
Page scrambling in m = 2 log N steps. (a) Themean deficit (cid:104) ∆ S (2) A (cid:105) from volume-law entanglement entropy,sampled over 2e4 random bipartitions A ∪ A of fixed size | A | ,decreases in the circuit E s on N = 128 qubits (solid red) at arate comparable to a random all-to-all circuit (dashed blue)and much faster than a comparable nearest-neighbor circuit(dotted green). (b) After 2 m circuit layers the mean Renyi en-tropy (cid:104) S (2) A (cid:105) (red diamonds), nearly saturates the Page curve(red), compared to a nearest-neighbor circuit of the samedepth (b, inset). (c) The mean entropy deficit (cid:104) ∆ S (2) A (cid:105) agreeswith random matrix theory (dotted black) to within samplingfluctuations for N = 16 , , , ,
256 (light to dark). (d)The fraction f (cid:15), | A | of subsystems A having less than maximalentanglement entropy (white dots) vanishes exponentially asa function of ∆ S (2) A / ln 2 = (cid:15) = 0 , , ,
3, in agreement withrandom matrix theory (vertical bars, light to dark). Errorbars shown or smaller than markers; lines are guides to theeye; gray windows show statistical noise floor. consisting of an extensive number | A | = N/ − t ∗ = 2 m = 14 interaction layers, as measuredby the Renyi entropy S (2) A ≡ − ln Tr (cid:2) ρ A (cid:3) of the reduceddensity matrix ρ A ≡ Tr A [ ρ ] (Fig. 2a). The averagedeficit (cid:104) ∆ S (2) A (cid:105) ≡ | A | ln 2 − (cid:104) S (2) A (cid:105) from perfect volume-law entanglement, sampled over 2 × randomly-chosenbipartitions A ∪ A (solid red), rapidly decreases as afunction of interaction layer t , saturating the Page limit∆ S (2) A = 2 | A |− N − (horizontal red) [1, 50] prior to layer t ∗ = 2 m . The timescale t ∗ ∼ log N required for completescrambling is comparable to that of a random all-to-allcircuit (dashed blue) – generally regarded as a prototyp-ical fast scrambler [2, 3, 8, 11, 51] – and much shorterthan for a nearest-neighbor circuit constructed withoutshuffling operations (dotted green).In fact the 2 m interaction layers of the circuit E s suf-fice to generate volume-law mean entanglement entropy (cid:104) S (2) A (cid:105) ≈ | A | ln 2 at all length scales | A | < N/ ρ = E s [ ρ ]. Randomly-chosen bipartitions A ∪ A , when organized by subsystem size | A | , reveal anearly ideal Page curve [1, 50] (Fig. 2b, red). The meanentanglement deficit (cid:104) ∆ S (2) A (cid:105) is extremely small for al-most all subsystem sizes and becomes substantial onlyfor very large | A | ∼ N/
2. Moreover, it is in excellentagreement with the predictions of random matrix theory(RMT) for binary matrices representing random stabi-lizer states over a range of system sizes (Fig. 2c) [45].The widespread delocalization of information gener-ated by the scrambling circuit E s is especially apparentwhen one considers how unlikely it is to find a subsystem A of the output state ρ with anything less than maxi-mal entanglement (Fig. 2d). Because the scrambling cir-cuit E s consists entirely of gates chosen from the Cliffordgroup, the Renyi entropy differs from its maximum valueonly by discrete bits ∆ S (2) A / ln 2 = (cid:15) = 0 , , , . . . [48, 49].We therefore count the fraction f (cid:15), | A | of the sampled bi-partitions whose Renyi entropies differ from maximal byan amount (cid:15) (Fig. 2d). We find that exponentially-manysubsystems A have maximal entanglement entropy (cid:15) = 0(for | A | < N/ A with entropy deficit (cid:15) > E s naturally serves a practical function in the context of FIG. 3.
Deterministic scrambling in the Hayden-Preskill thought experiment. (a) Scrambling in thecircuit E s can be characterized by the mutual information I (2)2 ( A : RB ) between Alice’s register A (red) and Bob’s reg-isters R, B (blue). (b) For N = 128 qubits, the mutual infor-mation grows rapidly as a function of Bob’s output register R over a range of message sizes | A | = 1 , , , , | R | min ≥ | A | + k ofoutput qubits with k ≤ | A | = 5, I (2)2 ( A : RB ) shows strong data collapse as a functionof system size N = 16 , , , ,
256 (light to dark). Errorbars smaller than markers; lines are guides to the eye. quantum error correction and quantum communication.In particular, strongly-scrambling quantum channels areknown to be excellent encoders that optimally protectquantum information against the effects of single-qubiterasure and other forms of local dissipation [9, 36, 51].While prototypical examples of such encoding circuitsare usually random, we demonstrate here that our de-terministic circuit E s can be leveraged for precisely thesame task, as illustrated by the thought experiment ofHayden and Preskill [9, 36–38] (Fig. 3). Here, quantuminformation held by a local observer Alice A is dumpedinto the strongly scrambling quantum channel E s and issubsequently recovered with high fidelity by a maximally-entangled observer Bob after measuring only a small sub-set R of the output qubits and neglecting the rest R . Highfidelity teleportation of Alice’s quantum information toBob’s register B occurs if and only if the unitary chan-nel is strongly scrambling [9, 37] and therefore presents asharp criterion for diagnosing the presence of scramblingdynamics in our circuit E s .From the perspective of quantum error correction, weview the scrambling circuit E s as an encoding circuit thatoptimally protects Alice’s information against erasure, al-lowing Bob to successfully reconstruct Alice’s state evenafter discarding the large majority of output qubits R .This is guaranteed in principle by large bipartite mutualinformation I (2)2 ( A : RB ) = S (2) A + S (2) RB − S (2) ARB (4)between the qubits A in Alice’s control and those R, B inBob’s control (Fig. 3a). Numerical calculations with Clif-ford circuits demonstrate that the circuit E s on N = 128qubits performs quite well as an encoding channel: themutual information increases linearly with the numberof output qubits | R | collected by Bob (Fig. 3b) andrapidly saturates to within 5% of its maximum value I (2)2 ( A : RB ) = 2 | A | ln 2 after he has collected a few morethan | A | qubits. Physically, this implies that Bob needonly gather a few | R | min ≥ | A | + k of the output qubitsin order to successfully decode Alice’s message (Fig. 3c),with k ≤ N . By contrast, nearest-neighbor cir-cuits of the same depth (Fig. 3b, dashed lines) show lowmutual information over a large range of output qubits | R | . For fixed message size | A | = 5, the mutual informa-tion shows strong data collapse as a function of systemsize N (Fig. 3d), indicating robustness to finite-size ef-fects.The numerical evidence presented in Figs. 2, 3 demon-strates that E s is a fast scrambler in the ideal unitary case.Any realistic implementation of this scrambling circuit,however, must contend with the effects of noise and dis-sipation that will inevitably degrade its performance. Inthe following, we analyze a possible experimental realiza-tion of E s in detail including the effects of decoherence tocharacterize its scrambling properties in a realistic setup.We propose to use long-lived ground states | (cid:105) , | (cid:105) ofneutral atoms as qubit states [29, 52, 53]. Single-qubit ro-tations allow for implementation of Hadamard and Phasegates. By exciting | (cid:105) to a Rydberg state, controlled- Z gates between neighboring atoms can be realized us-ing strong van der Waals interactions [20, 25–28, 54–56].Current experiments already achieve Rydberg gate fideli-ties > .
99 [20, 56]. A primary advantage of these op-erations is that they may be applied in parallel, usingglobal optical or RF pulses. For our simulations, we takeinto account crosstalk between atoms separated by thedistance r , resulting from the 1 /r -decay of the van derWaals interaction. We model decoherence as dephasingnoise with error rate p per atom after each interactionlayer [45].To distinguish between scrambling and decoherence,we attempt to recover Alice’s information using a proba-bilistic decoding circuit (Fig. 4a, dotted purple), follow-ing the scheme of Yoshida, Kitaev, and Yao [36–38]. Thisdecoder consists of a complex-conjugated copy of thescrambling circuit and the ability to measure EPR pairs;decoding protocols of this type have been realized in pio-neering experiment with trapped ions [57]. In the unitarycase p = 0, the circuit decodes Alice’s quantum informa-tion with a fidelity F EPR = 2 I (2)2 ( A : RB ) − | A | , conditionedon successful detection of | R | EPR pairs by Bob withprobability P EPR = 2 − I (2)2 ( A : RB ) (Fig. 4b). Bob’s abilityto recover Alice’s information is degraded by decoherence p >
0, where the product δ ≡ P EPR F EPR | A | ≤ δ behaves the same forthe nearest neighbor circuit and the scrambling circuit E s (Fig. 4d), for p >
0, the reachable teleportation fidelity F EPR is significantly smaller for the slow scrambling near-est neighbor circuit E s (Fig. 4c). This demonstrates thatfast scrambling is crucial in non-error-corrected systems,precisely because fewer gates provide fewer opportunitiesfor dissipation. Our scrambling circuit E s is optimal inthis regard as it generates strong scrambling using theminimal number of interaction layers 2 m ∼ O (log N ) al-lowed by the fast scrambling conjecture [2, 3, 8].We have shown how deterministic, highly-nonlocal it-erated (Floquet) circuits can generate fast scramblingdynamics in a way that is amenable to direct experi-mental realization using fast shuffle operations on neutralatom qubits. This technique allows for rapid long-rangespreading of entanglement while minimising errors fromexcitation of atoms to Rydberg states, and uses only shuf-fling operations, global single-qubit rotations and parallelnearest-neighbor interactions. Building fast scrambling FIG. 4.
Information scrambling in the presence of dis-sipation. (a) Scrambling in the circuit E s is diagnosed by thefidelity F EPR of recovering Alice’s quantum information A onBob’s register C using a probabilistic decoding circuit (dottedpurple). (b) At fixed circuit depth t = 6 the fidelity growswith the number of qubits | R | used in the decoder, indicat-ing successful teleportation of Alice’s information with fidelity >
50% even in the presence of single-qubit errors at rates p = 0 . , . , . . . , .
04 per two-qubit gate (light to dark). For p = 0 the fidelity is nearly identical to that of a Haar-randomcircuit (dotted black). (c) The fidelity (dots, solid lines) growswith circuit depth t , and substantially outperforms nearest-neighbor circuits of the same depth (crosses, dotted lines) inthe presence of dissipation. (d) The dissipation parameter δ falls as a function of circuit depth in both the scrambling cir-cuit and nearest-neighbor circuit. Each datapoint averagedover 6e4 quantum trajectories, with error bars smaller thanmarkers; lines are guides to the eye. circuits in the laboratory opens connections to a widerange of ongoing areas, including fundamental limits onthe spreading of quantum information [2, 3, 8, 58], ex-perimental studies of toy models of black holes [10, 59–62], efficient encoders for quantum error-correcting codes[9], and highly-entangled resources for quantum compu-tation [31, 32]. While we simulate example cases withstabilizer states [48, 49, 63] for large system sizes, anal-ogous circuits built in the laboratory may employ arbi-trary quantum rotations, exploring the complete many-body Hilbert space. We note that these graphs might alsobe constructed by other means, including collisional gateimplementations for neutral atoms, or via direct wiringof hypercubic coupling graphs in superconducting qubitsystems.In the final stages of this work, we became aware ofa proposal [62] for further explorations of many-bodyquantum teleportation, based around nearest-neighborRydberg models with scrambling times t ∗ ∝ N . Theprotocols we describe here for fast scrambling could beimmediately combined with these interesting proposalsto extend the example from Fig. 4 discussed here. ACKNOWLEDGMENTS
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DETERMINISTIC FAST SCRAMBLING WITH NEUTRAL ATOM ARRAYS:SUPPLEMENTAL MATERIALI. SHUFFLING OPERATIONS WITH OPTICAL TWEEZERS A Faro shuffle or perfect shuffle R (Fig. S1a) begins with N atoms (red dots, blue circles) labeled i = 0 , , . . . , N − x ( i ) = ia of a fixed 1D optical lattice (gray boxes) with spacing a . An additional N emptysites x = N a, . . . , (2 N − a are reserved for ‘scratch space.’ An auxiliary 1D tweezer array superimposed on the fixedlattice captures all N atoms and performs an adiabatic row-stretch operation that relocates atoms at site x to site 2 x (Fig. S1a(i)). The first N/ N/ x = − ( N − a ,and back into the trap. Upon switching off the tweezer array, the atoms i are rearranged in the linear trap by thepermutation i (cid:48) = R ( i ) (Fig. S1a(iii)). The inverse operation R − is executed by simply reversing the above steps.The simplest shuffling operation R deposits the atoms into the optical lattice with uniform density, leading totranslation-invariant nearest-neighbor Ising couplings in the subsequent interaction step. By slightly modifying theoriginal scheme the atoms may be deposited into the optical lattice non-uniformly such that subsequent Rydberg-Rydberg interactions generate controlled- Z gates only between nearest-neighbor pairs (Fig. S1b), while atoms sep-arated by more than one lattice site have negligible interactions. In particular, by using a rarified atomic arrayand preferentially shifting the transported atoms left or right by one lattice site at the end of the interleaving step(Fig. S1b(ii)), controlled- Z gates may be placed only on even (Fig. S1b(iii)) or odd (Fig. S1b(iv)) bonds of the ex-panded 1D lattice. These even- and odd- interaction layers, along with global Hadamard and Phase gates, can beused to readily construct ‘bricklayer’ circuits of non-commuting gates as shown in Fig. S1c.The shuffling operations described here are quasi-1D in that they maintain the one-dimensional linear geometryof the right-hand atoms (blue circles) during transport and utilize only straightforward spatial translations and row-stretch operations of the auxiliary tweezer array. These operations are straightforward to implement by sweepingthe frequencies of the tweezer sidebands in the independent RF drive signals f x , f z . The speed of these frequencysweeps and the accompanying spatial motion of the atoms will be limited by adiabaticity; recent experimental workhas demonstrated adiabatic transport speeds up to 75 µ m / ms, suggesting that single shuffling operations R can berealistically executed on timescales of order 1ms for N = 32 or more atoms in optical lattices with spacing a = 3 µ m[23]. FIG. S1.
Quasi-1D Shuffling Operations (a) Uniform-density shuffling procedure R for N = 16 atoms, similar to maintext. (b) Even- and odd-shifted shuffling procedures for N = 8 allow subsequent controlled- Z gates to be applied preferentiallyon even (iii) or odd (iv) bonds of the 1D lattice, respectively. (c) When combined with global single-qubit rotations (purple),even- and odd-bond gates (red) can be used to generate bricklayer circuits composed of non-commuting two-qubit gates (blue). II. GRAPH STATES
Given an undirected graph G = ( V, E ) with vertices i, j ∈ V and edges ( i, j ) ∈ E , a graph state is defined as | G (cid:105) = (cid:89) ( i,j ) ∈ E CZ ( i,j ) | + (cid:105) V (S1)where | + (cid:105) V = (cid:81) i ∈ V ( | (cid:105) + | (cid:105) ) / √ ( i,j ) is a controlled- Z gate applied between qubits i, j that live on the verticesof the graph [30]. This state is generated by preparing N qubits in | (cid:105) , applying a global Hadamard rotation, andapplying a controlled- Z gate between sites i, j for each edge ( i, j ) ∈ E . Because the CZ ( i,j ) gates mutually commute,the edges in the graph have no preferred ordering and every undirected graph G is in one-to-one correspondence witha graph state | G (cid:105) .The entanglement properties of the graph state | G (cid:105) can be immediately extracted from the adjacency matrix Γ ij of the graph G : Γ ij = (cid:40) i, j ) ∈ E A ∪ A , we may reorder therows and columns of Γ ij to bring it into block formΓ ij = (cid:20) Γ AA Γ AA (Γ AA ) T Γ AA (cid:21) (S3)where the off-diagonal sub-matrix Γ AA represents the edges connecting regions A, A . Then the Schmidt rank r ( ρ A )of the reduced density matrix is given by the binary rank (i.e. rank over the field GF(2)) of the sub-adjacency matrix[30] r ( ρ A ) = rank GF(2) (Γ AA ) (S4)From the Schmidt rank we can compute entropy measures such as the Renyi-2 entropy. In particular, for graph statesand stabilizer states the Renyi-2 entropy is simply S (2) A = − ln Tr (cid:2) ρ A (cid:3) = r ( ρ A ) ln 2.Graph states with sufficient nonlocal connectivity in the graph G may be Page entangled , meaning that they exhibitvolume-law entanglement S (2) A ∝ | A | in all bipartitions A ∪ A of size | A | < αN for some O (1) constant α [2, 3]. Asdiscussed above, volume-law entanglement in the reduced state ρ A is equivalent to the off-diagonal adjacency matrixΓ AA having full rank. The graph state | Q m (cid:105) , whose adjacency matrix is plotted in Fig. S2a, is Page-scrambled with α > ∼ / AA (Fig S2b). FIG. S2.
Hypercube graph state | Q m (cid:105) . (a) Adjacency matrix Γ ij for the hypercube graph Q m on N = 128 vertices. (b) 10 randomly-chosen subsystems A , arranged by size | A | , exhibit perfect volume-law entanglement (pink box) for all subsystemsup to | A | ≈ N/
8. For larger subsystems | A | > ∼ N/ A with entropy deficit (cid:15) > ,
2, but the averageentropy is still dominated by volume-law behavior. III. STABILIZER STATES
The idea of stabilizer formalism is to specify a state, stabilizer state , as a simultaneous eigenstate of a set of operatorscalled stabilizers . A state | Σ (cid:105) is said to be stabilized by a stabilizer O if | Σ (cid:105) is a +1 eigenstate of O as an action of O on | Σ (cid:105) does not change the state. Such a set of operator can be constructed by making use of the properties of Pauligroup on N qubits, P N . P N is a group consists of all the possible tensor products of the Pauli operators ( I , X , Y , Z )and phases (1, i , − − i ) For example, − I ⊗ X ⊗ Z ⊗ I ⊗ Y is an element, also referred to as a Pauli string , of a group P . An operator, O | Σ (cid:105) , which stabilizes and uniquely defines | Σ (cid:105) is constructed from a set of N linearly independentPauli strings, P N Σ ∈ P N , that stabilize the state | Σ (cid:105) , as follows O Σ = 12 N N (cid:89) P ∈P N Σ ( P + I ) , (S5)where the linear independence of Pauli string means a Pauli string P i ∈ P N Σ cannot be represented as a product of P j ( j (cid:54) = i ) and | Σ (cid:105) is a unique simultaneous eigenstate of the strings of P N Σ [48, 49, 64, 65]. A. Classical Simulation of Stabilizer States in Clifford Circuits
Consider a unitary operator O C which transforms an element of Pauli group to some element in the same group.The action of such unitary on a stabilizer state | Σ (cid:105) produces a stabilizer state | Σ (cid:48) (cid:105) , which is stabilized by an operator O Σ (cid:48) = O C O Σ O † C . A set of operators {O C } ≡ C N with such properties forms Clifford group on N qubits. The evolutionof a stabilizer state under actions of the elements of the Clifford group can therefore be simulated by keeping track ofhow N Pauli strings in the initial stabilizer are being mapped.A stabilizer state undergoing unitary evolution by the elements in the Clifford group known to be able to computeclassically in polynomial time as proven by Gottesman and Knill[48, 49]. The simulation of the stabilizer states inClifford circuit can be done efficiently with a logical operation on a N by 2 N + 1 binary matrix M [49]. In this binarymatrix, each row represents a Pauli String. For each row, l th character of the string is encoded by mapping thenumber of X to l th column and the number of Z to N + l th column. Consequently, having Y = iXZ corresponds to1 , l th and N + l th columns. The last column is reserved for tracking the overall phase of the element, where 0corresponds to −
1, and 1 corresponds to +1. As Clifford gate transforms one Pauli string to another, evolution of astabilizer state | Σ (cid:105) can be kept track by altering the strings of initial P N Σ accordingly to the transformation rules[49].It is known that any N -qubit Clifford group can be generated from combinations of Hadamard ( H ), Phase ( P ) andControlled-NOT (C-NOT) gates [48, 49, 65] acting on different sets of qubits in the system. Hadamard gate actingon a site maps operators Z to X and X to Z in a Pauli string. Phase gate acting on a site, on the other hand, maps X to Y and Y to X . C-NOT gate, acting on two qubits, control qubit l and target qubit m , flips the target basiswhenever the state of the l is | (cid:105) , i.e.C-NOT l,m = 12 (( I l − Z l ) + ( I l + Z l ) X m )) . (S6)By computing the unitary transformation on all the element in the 2-qubit Pauli group, one finds that there are only4 non-trivial transformation rules: X l I m to X l X m , I l X m to I l X m , Z l I m to Z l I m , I l Z m to Z l Z m . (S7) B. Construction of Random Nearest Neighbor and Random All-to-All Models
Two random circuits that are investigated in this paper in accordance to Fig. 2a are random nearest-neighbor circuitand random all-to-all curcuit. The random nearest-neighbor circuit is simulated by acting randomly chosen gates fromthe 2-qubit Clifford group (including single site rotations and identities) on even bonds at even interaction layers,and odd bonds at odd interaction layers, making a brickwork pattern. Random all-to-all circuit, on the other hand,for each interaction layer, the randomly chosen gates are applied to the even bonds and after the parallel applicationof N/ N/ N/ C. Entanglement Entropy of Stabilizer States
In Sec. III. A, the stabilizer state | Σ (cid:105) is defined with a set of N linearly independent Pauli strings { P i } whichstabilizes | Σ (cid:105) and | Σ (cid:105) being the unique simultaneous eigenstate of the strings. The uniqueness of | Σ (cid:105) and the propertywhich P i having only two, namely +1 and − O Σ = N (cid:81) P ∈P N Σ ( P + I ) (Eq. (S5)) is a projector that projects onto asubspace | Σ (cid:105) (cid:104) Σ | , or equivalently it is the density matrix of the state | Σ (cid:105) . Knowing the density matrix, entanglemententropy of any subsystem A of | Σ (cid:105) can be calculated readily[51, 66]. Expanding the products in O Σ , O Σ = 12 N (cid:88) g ∈G g, (S8)where a set G is a set generated by all the possible products of { P i } and an identity I ⊗ N , O can be written as a sumof Pauli strings as Pauli Group is closed in multiplication. The reduced density matrix on the subsystem A , ρ A isobtained by tracing out the complement of A , A . This is equivalent to throwing away the Pauli strings which theparts corresponding to A is not an identity as Pauli matrices are traceless except for the identity ρ A = Tr A [ ρ ] = 2 | A | N (cid:88) g A ∈G A g A = 12 | A | (cid:88) g A ∈G A g A , (S9)where G A ∈ G is a set of all g with Tr A { g } (cid:54) = 0[66]. Let N A be the number of linearly independent Pauli strings thatgenerate g A , then (cid:80) g A ∈G A g A is proportional to a projector of rank 2 | A |− N A as this projector projects out the − S A is S A = ( | A | − N A ) ln 2 (S10)Using 2 N A + N A = 2 N and equivalence of N A to the binary rank of M A , where M is the binary matrix representingthe state | Σ (cid:105) and M A is the matrix with columns of M corresponding to the subsystem A , S A = (cid:0) rank GF(2) ( M A ) − | A | (cid:1) ln 2 . (S11)From S A = S A , we obtain S A = (cid:0) rank GF(2) ( M A ) − | A | (cid:1) ln 2 . (S12) D. Average Entropy of a Subsystem of Random Stabilizer States
As it is shown in the previous section, a stabilizer state of N qubits can be represented by a binary matrix M of the dimensions N by 2 N . A random stabilizer state can therefore be constructed from a random binary matrixwith a constraint rank GF(2) ( M ) = N . Also it is shown that the entropy of a subsystem of size | A | of a stabilizerstate can be obtained by subtracting | A | from the binary rank of the binary matrix of the corresponding region. Theaverage entropy of a random subsystem A in a random stabilizer state can therefore be estimated by approximatingthe rank GF(2) ( M A ) with that of random binary matrices.An N by 2 | A | (2 | A | < N ) random binary matrix M | A | can be constructed by appending 2 | A | − N by 1 matrix. Each time a new row is added, the rank does not increase with probability 2 k / | A | where k is the current rank, and the rank increases otherwise. The probability of the matrix to have exactly r whenthe full matrix is constructed is, therefore, P (rank GF(2) ( M | A | ) = r ) = (cid:88) t ∈T r (cid:89) i =1 (cid:18) i − | A | (cid:19) t i − t i − − (cid:18) − i − | A | (cid:19) (S13)where T is a set of all the configurations of the row numbers where rank increases by 1 and for all t ∈ T , t = 0. Forlarge | N | , the above expression can be approximated by the following expression [67] P (cid:0) rank GF(2) ( M | A | ) = 2 | A | − (cid:15) (cid:1) ≈ − (cid:15) ( N − | A | + (cid:15) ) × ∞ (cid:89) i = (cid:15) +1 (cid:18) − i (cid:19) N − | A | + (cid:15) (cid:89) i =1 (cid:18) − i (cid:19) − . (S14)2 FIG. S3. (a) The entanglement entropy of a region consists of r consecutive qubits starting from site i , S i ( r ), for i = 0and i = N/ N = 128 at the number ofinteractions t = 2 log ( N ) = 14. The bar on ¯ S i ( r ) indicates that it is an averaged quantity over the different realizations ofthe random circuit. For this simulation the average of up to 1000 realizations are taken. (b) The average entropy of up to2e4 random subsystems, A , of an output state of a single trajectory of the random nearest-neighbor circuit ordered by thesubsystem size | A | (green dotted line). The theory line (red solid line) is computed using Eq. (S16) with the entropy as functionof r consecutive regions approximated by S ( r ) ∼ ˜ S ( r ) = (cid:0) ¯ S ( r ) + ¯ S N/ ( r ) (cid:1) . Here the average of the functions ¯ S ( r ) and¯ S N/ ( r ) are taken to take account of the effect from the open boundary condition. Using this expression, the average entropy deficit of a subsystem of size | A | of a random stabilizer state is approximatedas follows (cid:104) ∆ S (2) A (cid:105) = (cid:32)(cid:88) (cid:15) (cid:15)P (cid:0) rank GF(2) ( M | A | ) = 2 | A | − (cid:15) (cid:1)(cid:33) ln 2 ≈ | A |− N ln 2 (S15)where the approximation is made by only considering (cid:15) = 0 and 1, and taking the limit of 1 (cid:28) N − | A | . This resultcoincides with but slightly larger than the expected entropy deficit of a Haar random state[1], by a constant factorof 2 ln 2, which is of the order 1. Also, a random binary matrix constructed in this way is almost guaranteed to havethe rank of N for 1 (cid:28) N as the probability of the binary rank of the matrix to be N − (cid:15) with (cid:15) = 1 is 2 − N . Thus thevast majority of random N × N binary matrices represent stabilizer states. IV. AVERAGE ENTROPY OF A SUBSYSTEM OF AREA-LAW STATES IN 1-D QUANTUM SYSTEMS
In this section, we derive the expression for average entropy of a subsystems A consisting of randomly chosenqubits with the subsystem size of | A | drawn from the N -qubit state with area-law entanglement entropy. Let theentanglement entropy of r consecutive region from i th qubit to be expressed as S i ( r ). We assume the translationalsymmetry of this function such that S i ( r ) = S j ( r ) = S ( r ) also holds for sites j (cid:54) = i . Given a particular subsystem andconfiguration A ∈ A , one can always find Q A sets of q A,k ( k = 1 , , . . . , Q A ) qubits drawn from a consecutive region inthe system. For example a set A = { , , , , } has Q A = 3 with q A, = 1 (from { } ), q A, = 3 (from { , , } ), and q A, = 1 (from { } ). Assuming that the mutual information between the two sets are 0, which is true for the vastmajority of cases for the state with the area-law entanglement entropy for | A | (cid:28) N/
2, one can write the entanglemententropy of a given configuration as (cid:80) Q A k =1 S ( q A,k ). For fixed | A | and Q A , there are C ( Q A ) = (cid:18) N − | A | + 1 Q A (cid:19) possibleways to draw | A | qubits from N qubits such that they have exactly Q A sets of { q A,k } consecutive regions. Finallythere are p ( | A | ) = Q A ways to partition a subsystem A into the cells which contains at least 1 consecutive qubits,where p ( | A | ) is a well known function in number theory called partition function (not to be confused with the partitionfunction from the thermodynamics) [68]. From these, the expression for the average entropy of a random subsystemof size | A | of an area-law entangled quantum state is given as (cid:104) S A (cid:105) = (cid:80) p ( | A | ) l =1 C ( Q A l ) (cid:80) Q Al k =1 S ( q A l ,k ) (cid:80) p ( | A | ) l =1 (cid:80) Q Al k =1 C ( Q A l ) (S16)where A l ( l = 1 , , . . . , p ( | A | )) goes through all the different ways to partition the subsystems. As shown in Fig. S3b,the average entropy of the random bipartitions as a function of the subsystem size result of the numerical simulation of3random nearest-neighbor circuit for N = 128 at t ∗ = 2 m = 14 shows an excellent agreement with the theoretical valuescomputed explicitly with Eq. S16 with S ( r ) = ˜ S ( r ) = (cid:0) ¯ S ( r ) + ¯ S N/ ( r ) (cid:1) where ¯ S ( r ) and ¯ S N/ ( r ) are estimated byaveraging up to 1 × realizations of the circuit (Fig. S3a). V. CHARACTERIZING SCRAMBLING VIA THE HAYDEN-PRESKILL EXPERIMENT
Originally conceived in the context of the black hole information problem, the Hayden-Preskill thought experiment(Fig. 3) can be viewed as a general conceptual tool useful for characterizing the scrambling properties of quantumchannels E . We focus for the moment on unitary channels E = U and consider a local observer Alice who wishesto use this channel to encode some quantum information. To do so, she maximally entangles her qubits A with thechannel U using | A | Bell pairs | EPR (cid:105) A = (cid:81) | A | ( | (cid:105) + | (cid:105) ) / √ A , Alice has complete control over the information entering the channel U ; in particular, to send a quantumstate | ψ (cid:105) into the encoding circuit U , Alice need only project her maximally-entangled qubits A onto the desired state | ψ (cid:105) .Bob, another observer whose qubits B are maximally-entangled with the remaining channel inputs, attempts torecover Alice’s quantum information by collecting a subset R of the output qubits. Of course if Bob has access to allof the output qubits | R | = N then his ability to reconstruct Alice’s information is trivially guaranteed by the unitarityof U . Surprisingly, however, if the operator U is strongly scrambling then Bob can reconstruct Alice’s informationusing only a handful | R | min = | A | + k of output qubits, with k an O (1) constant independent of | R | , | A | , N . We cansee why by examining the information content available to the regions A, B, R, R .The EPR pairs held by Alice and Bob in regions
A, B serve to convert the operator U into a pure state | U (cid:105) via thechannel-state correspondence | U (cid:105) = U | EPR (cid:105) [4] as illustrated in Fig. 3a. We may therefore use entropy measures onthe state | U (cid:105) to quantify the amount of information that is accessible to various observers. In particular, the mutualinformation I ( A : RB ) = S A + S RB − S ARB = S A + S AR − S R = 2 | A | ln 2 − I ( A : R ) (S17)quantifies the amount of information shared between the region A in Alice’s control and the regions R, B in Bob’s(we have dropped the superscripts S (2) , I (2)2 for notational simplicity). In the last two lines we have used the factthat S ARB = S R and S RB = S AR which follow from the unitarity of U . Bob’s ability to reconstruct Alice’s quantuminformation is guaranteed in principle by maximal mutual information I ( A : RB ) ≈ | A | ln 2 between region A andregion R ∪ B [9, 36, 37]. As shown in Eq. (S17), this is equivalent to having vanishing mutual information I ( A : R )between A and R such that the output qubits R alone reveal nothing about Alice’s information. If this is the case thenBob can afford to ignore (or erase) the region R and still recover Alice’s information, so long as he maintains controlover regions R, B . In this language, the relation of scrambling to quantum error correction becomes particularly clear:vanishing mutual information I ( A : R ) implies that Alice’s quantum information is protected from all errors, up toand including erasure, acting on the qubits R . VI. DETAILS OF NUMERICAL SIMULATIONSA. Page Scrambling of Polarized States in Different Basis
In this section, we show that the circuit E s can scramble a state regardless of the basis that is chosen for the initialstate. In Fig. 2, the initial state of | (cid:105) ⊗ N , where N is the system size, is computed to provide the various numericalevidence of the scrambling nature of the circuit. Shown in Fig S4 are the average entropy deficits of random subsystems A of the x , y , and z -polarized initial states, which is the same quantity computed in Fig. 2c of the main text for the z -polarized initial state. As it can clearly be seen in the figure, the output states at the number of interaction layers t = 2 m saturates the Page limit, showing that the circuit scrambles regardless of the choices of the basis of the initialstates.4 FIG. S4. Average entropy deficits, (cid:104) ∆ S (2) A (cid:105) / ln 2, of the x (red) , y (green), and z -polarized (blue) initial states for up to 2e4random subsystems A compared to the Page limit (black dashed) computed from the Random Matrix Theory of GF(2) for thesystem sizes N = 16 , , , ,
256 (light to dark) at t ∗ = 2 m = 2 log ( N ). The points that are not shown for certain valuesof | A | , because configurations with the entropy deficit of larger than 0 could not be found due to the low occurrence of suchconfigurations. B. Simulating Probabilistic Decoding with Quantum Trajectories