PHYSICAL REVIEW X 10, 041020 (2020)
Dynamical Purification Phase Transition Induced by Quantum Measurements
Michael J. Gullans 1 and David A. Huse 1,2 1Department of Physics, Princeton University, Princeton, New Jersey 08544, USA 2Institute for Advanced Study, Princeton, New Jersey 08540, USA
(Received 23 September 2019; revised 3 July 2020; accepted 22 September 2020; published 28 October 2020)
Continuously monitoring the environment of a quantum many-body system reduces the entropy of (purifies) the reduced density matrix of the system, conditional on the outcomes of the measurements. We show that, for mixed initial states, a balanced competition between measurements and entangling interactions within the system can result in a dynamical purification phase transition between (i) a phase that locally purifies at a constant system-size-independent rate and (ii) a “mixed” phase where the purification time diverges exponentially in the system size. The residual entropy density in the mixed phase implies the existence of a quantum error-protected subspace, where quantum information is reliably encoded against the future nonunitary evolution of the system. We show that these codes are of potential relevance to fault-tolerant quantum computation as they are often highly degenerate and satisfy optimal trade-offs between encoded information densities and error thresholds. In spatially local models in 1 þ 1 dimensions, this phase transition for mixed initial states occurs concurrently with a recently identified class of entanglement phase transitions for pure initial states. The purification transition studied here also generalizes to systems with long-range interactions, where conventional notions of entanglement transitions have to be reformulated. We numerically explore this transition for monitored random quantum circuits in 1 þ 1 dimensions and all-to-all models. Unlike in pure initial states, the mutual information of an initially completely mixed state in 1 þ 1 dimensions grows sublinearly in time due to the formation of the error-protected subspace. Purification dynamics is likely a more robust probe of the transition in experiments, where imperfections generically reduce entanglement and drive the system towards mixed states. We describe the motivations for studying this novel class of nonequilibrium quantum dynamics in the context of advanced quantum computing platforms and fault-tolerant quantum computation.
DOI: 10.1103/PhysRevX.10.041020 Subject Areas: Condensed Matter Physics, Quantum Physics, Statistical Physics
I. INTRODUCTION finite-temperature bath (i.e., an open quantum system) can be driven to a pure state remains less understood [6,7]. In thermodynamic equilibrium, pure quantum states can An essential resource in controlling open quantum only be achieved at absolute zero temperature. The non- systems is the ability to make measurements of the system, equilibrium thermodynamic cost of purification is encoded which can then be used to perform feedback and conditional in the third law of thermodynamics, which states that it is control (e.g., the famous Maxwell demon) [8]. However, impossible to reach a zero entropy (pure) quantum state in a purification does not require any feedback because the finite amount of time. In quantum information science, continuous monitoring of the environment can be used to purification plays an essential role in many models of continually gain information about the system, thereby quantum computation, where one often assumes access to reducing the number of accessible states consistent with highly pure computational or ancilla qubits [1]. Although it the measurement record and the intermediate dynamics is known that the requisite purification is possible given [9–11]. Naively, one expects that continuous, perfect mon- sufficiently fine control over a quantum system and its itoring will rapidly purify the system; however, it is known environment [2–5], the question of whether a generic from the study of quantum-error-correcting codes that quan- interacting many-body quantum system coupled to a tum states can be protected from extensive numbers of local measurements [12–14]. Recently, there has been significant experimental progress towards realizing the requisite ingre- Published by the American Physical Society under the terms of dients for such measurement-driven purification of many- the Creative Commons Attribution 4.0 International license. – Further distribution of this work must maintain attribution to body states in quantum computing platforms [15 24]. the author(s) and the published article’s title, journal citation, In this article, we show that there is a dynamical purifi- and DOI. cation phase transition as one changes the measurement
2160-3308=20=10(4)=041020(28) 041020-1 Published by the American Physical Society MICHAEL J. GULLANS and DAVID A. HUSE PHYS. REV. X 10, 041020 (2020) rate in a class of random quantum circuit models with mea- systems have no notion of spatial locality or a well-defined surements. For pure initial states, it was shown recently that geometry, which requires one to revisit the conventional there is an entanglement phase transition in these models definition of an area-to-volume-law entanglement transi- from area-law to volume-law entanglement [25–27], with tion. Interestingly, however, the purification transition we subsequent work deepening our understanding of this find naturally persists in all-to-all coupled models with family of entanglement phase transitions [28–30].We k-local interactions. show that, for mixed initial states, this entanglement phase Because of the difficulty in isolating and measuring transition transforms into a dynamical purification transi- volume-law entanglement, purification dynamics can also tion between a “pure” phase, with a constant purification serve as a more robust probe of measurement-induced rate in the thermodynamic limit, and a “mixed” phase, entanglement transitions in experiments. For example, we where the purification time diverges exponentially in recently found that the purification dynamics of a finite system size. Thus, if one takes the simultaneous limit of number of qubits acts as a local order parameter [40], an infinite system and infinite time, with any power-law which, together with the quantum Fisher information [41], relation between system size and time, then an initially allows direct experimental access to the phase transition on maximally mixed state has a nonzero long-time entropy near-term quantum computing devices. We argue that an density in the mixed phase, while it becomes pure, and important motivation for studying this class of nonequili- area-law entangled, in the pure phase. brium quantum dynamics is to develop more efficient We provide a more general definition of purification routes to fault-tolerant quantum computation. transitions in terms of a phase transition in the quantum channel capacity of the underlying open system dynamics. II. OVERVIEW This definition can be applied to arbitrary quantum chan- nels, and it implies that a purification transition should be Before describing our analysis, we present an overview fundamentally interpreted as a type of quantum-error- of the main results on purification transitions and dynami- correction threshold. Therefore, our results help us further cally generated codes. Our definition of purification phase establish the connections between measurement-induced transitions in open system dynamics fundamentally relies phase transitions, channel capacities, and quantum error on a quantity known as the quantum channel capacity, correction [30]. We additionally strengthen these connec- which determines the maximum amount of quantum tions by showing that the unitary-measurement dynamics information that can be transmitted by a noisy quantum projects the system into an optimal quantum-error- channel [42–45]. A closely related quantity is the coherent correcting code space. This encoding achieves the capacity quantum information, which plays an important role in the for the future evolution of the channel in single-use error basic theory of quantum error correction [46–48]. correction. Thus, our work points to a large family of In Fig. 1(a), we provide a qualitative phase diagram for a previously unexplored codes with an optimal trade-off purification transition as a function of a generalized between code rates and error thresholds, with relatively measurement or decoherence rate p.Forp>pc, there is simple encoding schemes, that may provide useful insight no pair of encoding and decoding operations that can or applications to fault-tolerant quantum computation. protect an extensive amount of quantum information. Thus, To develop the basic phenomenology of purification the system always forgets initial conditions (purifies), and phase transitions, we numerically explore the measure- the dynamics is fundamentally irreversible. On the other ment-induced transition in the 1 þ 1-dimensional stabilizer hand, for p 041020-2 DYNAMICAL PURIFICATION PHASE TRANSITION INDUCED … PHYS. REV. X 10, 041020 (2020) (a) (b) Random Irreversible dynamics fascinating and rich problem, with potential applications encoding (Decoding fails) Decoding in fault-tolerant quantum computation. In Sec. VIII,we discuss a variety of motivations to better characterize these phases and the associated quantum-error-correcting Reversible (c) R codes in the ordered phase, as well as to investigate these dynamics Decoding dynamics in intermediate-scale quantum devices. After (Decoding succeeds) S reading Secs. III A and III B, those readers less interested Encoded inf. density Encoding M E in the proofs and further details of our analysis can avoid the intervening sections and Appendixes, or return to FIG. 1. (a) Qualitative phase diagram for a purification phase them later. transition. Inside the blue region it is possible to encode an The paper is organized as follows: In Sec. III,we extensive amount of quantum information and recover the initial introduce the random circuit model studied in this work state for all polynomial times in the thermodynamic limit with and establish the basic phenomenology of purification fidelity arbitrarily close to 1. Outside this boundary, there is no transitions in unitary-measurement dynamics. We then encoding and decoding pair that succeeds, and the dynamics is provide a more general definition of a purification tran- fundamentally irreversible. (b) Basic setup of the encoding- sition in terms of the long-time scaling of the quantum decoding problem for the class of channels studied in this work where the capacity-achieving encoding is implemented by the channel capacity with system parameters in the thermody- dynamics of the channel itself. (c) General model for open system namic limit. To make more explicit connections to unitary- dynamics using a reference R system and an environment E to measurement models, we introduce the notion of strong purify a quantum channel N . In the language of quantum error purification transitions, unitary-dephasing channels, and correction, the initial-state preparation is the encoding operation, monitored channels. In Sec. IV, we show rigorously that, while subsequent measurements and control are the decoding for monitored channels with a strong purification transition, operation. the late-time density matrix defines a quantum-error- correcting code space that can saturate the channel capacity Furthermore, we rigorously prove a sufficient set of bound for the future evolution of the system, with poten- conditions under which the monitored channel dynamics tially useful properties for fault-tolerant quantum compu- can act as an optimal encoding for the future evolution of tation. Furthermore, we show that the dynamics in the the system. We show numerically that these conditions are ordered phase has interesting parallels to more conven- satisfied for a representative family of models that exhibit tional decoding problems in quantum error correction. In measurement-induced entanglement transitions. The associ- Sec. V, we discuss the class of monitored random circuit ated high-fidelity recovery operations apply to a single use of models that have been broadly found to exhibit measure- the channel. Thus, the system in the mixed phase generates a ment-induced entanglement transitions, which we study in family of quantum-error-correcting codes that saturate funda- this work. We also describe the relation between quantum mental bounds on the trade-off between the density of encoded trajectories and our definition of a purification transition. In quantum information and the associated error threshold. Sec. VI, we present a detailed overview of the critical The basic concept underlying our theorem is illustrated properties of the entanglement and purification transition in in Fig. 1(b). We first run a monitored channel (defined 1 þ 1 dimensions based on a finite-size scaling analysis of below) for a particular mixed-state input ρ0 on a timescale numerical simulations. polynomial in the system size. The repeated rounds of In Sec. VII, we introduce a family of two-local, infinite measurements, followed by unitary scrambling dynamics, range models that display clear signatures of the purifica- randomly projects the system into a quantum-error- tion transition. We find that the transition in this model p correcting code-space density matrix ρm. Remarkably, this violates a version of the Hamming bound on c [49], encoding process is successful for the future evolution of indicating that the zero-rate codes near the critical point the channel that is statistically independent of the past are highly degenerate. Degenerate, zero-rate quantum evolution. Stated more precisely, a random “code word” codes, such as the surface code [50], can have high- ρ jPcii drawn from any ensemble representation of m ¼ error-correction thresholds and resilience to errors in gates λ and measurements, making them potentially useful for i ijciihcij can be approximately recovered back to this state following further evolution with the channel. For a fault-tolerant quantum computation. In Sec. VIII,we correct choice of ρ0, this encoding scheme protects the discuss the general phase diagram and universality classes maximal possible amount of quantum information in the of measurement-induced transitions opened up by the thermodynamic limit. These results have broad implica- purification perspective presented here. We then discuss tions for the study of measurement-induced phase tran- the implications of our work for fault-tolerant quantum sitions, as we describe in this article. computation and experimental studies of measurement- The classification of the complete family of information- induced phases in intermediate-scale quantum devices. We theoretic phases of unitary-measurement dynamics is a present our conclusions in Sec. IX. 041020-3 MICHAEL J. GULLANS and DAVID A. HUSE PHYS. REV. X 10, 041020 (2020) In Appendix A, we present a formal statement and proof of our theorem on dynamically generated codes. In Appendix B, we derive a bound on the channel capacity by the average entropy of unravelings of the channel into quantum trajectories. In Appendixes C–E, we provide more details on the stabilizer formalism, mixed stabilizer states, and methods to compute entropies of stabilizer states. In Appendix F, we introduce a quantity that can be useful in characterizing the optimal codes generated by the dynamics that we define as the contiguous code length. III. PURIFICATION TRANSITIONS In this section, we first present an overview of the phenomenology of purification transitions in the random circuit models studied in this work, which are based on a special class of quantum circuit dynamics known as stabilizer circuits that can be classically simulated in polynomial time. Such models were introduced in the context of measurement-induced entanglement transitions by Li, Chen, and Fisher [25,28]. We then provide a general definition of purification transitions in terms of the scaling of the channel capacity in the thermodynamic limit. This definition is essentially a reformulation of a quantum-error- correction threshold, but introducing it in this setting allows us to formulate the necessary concepts and terminology FIG. 2. (a) Random quantum circuit model studied in this work, starting from basic concepts in open quantum systems. To called the “random Clifford” model [25]. Local two-qubit make more concrete connections to unitary-measurement unitaries are drawn uniformly from the Clifford group in a 1D models, we introduce the notion of strong purification brickwork arrangement with periodic boundary conditions. transitions, unitary-dephasing channels, and monitored Between each layer of gates, projective Z measurements happen channels. Such channels satisfy a weaker version of the with probability p at each site. (b) Phase diagram for the late- Knill-Laflamme conditions for quantum error correction time, circuit-averaged entropy density hSðρÞi=L starting from [1,51] and form the basis for our theorem on dynamically the completely mixed initial state. We took time t ¼ 4L and 256 generated quantum-error-correcting codes. L ¼ and 512 to limit finite-size effects, although signatures of the transition appear already at L ¼ 8. The blue curve shows ν Aðpc − pÞ for A ≈ 7.3, p ≤ pc ¼ 0.1593ð5Þ and ν ¼ 1.28ð2Þ A. Random Clifford model obtained below. Measurement-induced phase transitions in ensembles of quantum trajectories are now understood to generically computational zero state or any stabilizer state, can be appear whenever there is a balanced competition between simulated on a classical computer in a time that scales unitary dynamics that increases the entanglement of the polynomially in L [52,53]. As a result, one can perform a system and a measurement process that reduces the finite-size scaling analysis of the transition for hundreds or entanglement. Therefore, to establish the connection to even thousands of qubits. Furthermore, polynomial-time purification transitions, we study the “random Clifford” classical algorithms have been introduced to compute model introduced in Ref. [28] [see Fig. 2(a)], where the entropies and mutual information of stabilizer states properties of the entanglement transition have been most [54,55]. In Appendixes C–E, we provide a more detailed well established due to the ability to perform large-scale overview of the formalism used to describe stabilizer numerics. circuits and states. The model consists of a “brickwork” circuit of random Although these special properties of stabilizer circuits two-site unitaries drawn uniformly from the Clifford group, make many aspects of their dynamics nongeneric, one of operating on a linear chain of L qubits with periodic the defining features of the Clifford group is that it forms a boundary conditions. In between each layer of unitaries, unitary t design for t ≤ 3 [56,57]. This property implies that each site is measured in the Z basis with a fixed probability channel-averaged properties of Clifford models often have p. One tunes through the phase transition by changing p. similar phenomenology to more general quantum chaotic Including the measurements, this random circuit is an models [55,58,59]. The initial state can either be a mixed or example of a stabilizer circuit, which, starting from the pure stabilizer state, where a mixed stabilizer state is 041020-4 DYNAMICAL PURIFICATION PHASE TRANSITION INDUCED … PHYS. REV. X 10, 041020 (2020) defined as the uniform mixture of all pure stabilizer states features of the mixed phase already appear when the associated with a given stabilizer group (see Appendix D). measurements occur only at a single given site at an A mixed stabilizer state with rank 2k has an equivalent arbitrarily slow rate (technically, a sufficient condition interpretation as a projector onto an ½N;k stabilizer code, for large L is p ≪ 1=L3 [60]). In this limit, the spatial where k refers to the number of logical, or encoded, qubits structure of the circuit is largely irrelevant for the late-time in the code. dynamics, and we can replace the unitary between mea- surements by a random Clifford gate that acts on the entire B. Purification phases set of L qubits. Starting from the completely mixed initial We now describe the basic signatures of the phases in the state, the density matrix after the first measurement is purification transition. One key signature of the transition is 1 shown in Fig. 2(b). Here, we take L ¼ 256 and 512 sites ρ1 I m1Z ; 1 ¼ 2L ð þ Þ ð Þ and, starting from the completely mixed initial state, ρ I=2L, we run many realizations of the random circuit ¼ where m1 ¼ 1 is the first measurement outcome, and Z is out to a time t (= number of two-site unitaries that have the Pauli-Z matrix on the site measured. The purity of the acted on each qubit) that is a fixed multiple of L. We then system has increased by a factor of 2. Following the ρ compute the entropy density of the resulting state hSð Þi=L intermediate-time dynamics and second measurement, averaged over random circuit realizations, assuming perfect the density matrix is updated as knowledge of the outcome of all measurements in each run of the circuit [25–27]. For each such stabilizer circuit that † P2U2ρ1U2P2 1 † starts from this completely mixed initial state, the von ρ2 ¼ ¼ ðI þ m2Z þ 2m1P2U2ZU P2Þ; † ρ 2L 2 Neumann and all Renyi entropies are equal at a given time, Tr½U2P2U2 1 although they differ between circuits and can decrease with ð2Þ time. Below a critical value of p ¼ pc ¼ 0.1593ð5Þ (deter- 1 mined to this level of precision below), we see that the late- where Pn ¼ 2 ðI þ mnZÞ is a projector onto the state time density matrix has residual entropy density that is consistent with the outcome of measurement n of m ¼ ρ ∼ − ν n independent of L with the scaling hSð Þi=L ðpc pÞ 1 and U is the unitary for the random circuit between ν 1 28 2 n for ¼ . ð Þ. In Sec. V, we show that this residual measurement n − 1 and n. To compute the denominator of entropy density directly implies an extensive channel † Eq. (2), we assume that U2ZU2 ≠ Z (this is true with capacity (see Sec. III C for a definition). Although we 1 probability 1 − since U2 maps Z to a random traceless cannot run the dynamics for these sizes out to exponentially 4L−1 product of Pauli operators on all L qubits), such that long times, our finite-size scaling analysis in Sec. VI is † consistent with an exponentially divergent lifetime of the P2U2ZU2 has zero trace. Using the property that the plateau value of SðρÞ. Clifford group is a 2-design, we can compute the circuit- On the other hand, for p>pc, the average entropy averaged purity after this second measurement, density decays to zero with a decay rate that is independent 2 L of L (leading to a purification time of about ln L). To the hTr½ρ2 i ¼ 3=2 : ð3Þ level of precision we can test, the values of pc and ν for the purification transition are identical to those for the entan- Extending these arguments to many subsequent measure- glement phase transition for pure initial states. In Sec. VI, ments, one finds that each time a measurement occurs, it we provide a detailed overview of the properties of the increases the average purity by only 1=2L,so system near the critical point. We now give some intuition 2 L L for the physical origin of the two purification phases deep hTr½ρn i ¼ ðn þ 1Þ=2 ;n≪ 2 ; ð4Þ in their respective regimes. The basic origin of the pure phase can be simply leading to an exponentially long purification time. The understood for p sufficiently close to 1. In this limit, each average entropy satisfies the inequality layer of measurements projects the system into a near 2 perfect product state in the Z basis. As a result, any hSðρnÞi ≥−loghTr½ρn i ¼ L − logðn þ 1Þ: ð5Þ correlations and complexity in the system can build up only over a few sites before being decohered by the This limiting case establishes some essential features of the measurements, which makes the system highly insensitive measurement-induced dynamics in the mixed phase, to initial conditions. Since pure states are a fixed manifold including the insensitivity of the basic phenomenology of the dynamics, the system will rapidly converge to zero to spatial locality. Although some aspects of the above entropy density, regardless of initial conditions. argument extend to small but L-independent p, to establish The mixed phase generally has a richer many-body the existence of the mixed phase in the present work, we dynamics than the pure phase. It turns out that some basic rely on numerical solutions of the model. In addition to 041020-5 MICHAEL J. GULLANS and DAVID A. HUSE PHYS. REV. X 10, 041020 (2020) 1 1 þ -dimensional models, we also study the mixed phase ρRE0 ¼ ρR ⊗ ρE0 ; ð9Þ in all-to-all models in Sec. VII. where ρA ¼ TrAc ½ρ is the reduced density matrix on A and c C. Purification transitions: Definitions A is the complement of A. This factorization implies the existence of a Schmidt decomposition for the pure state of In this section, we present a mathematical definition of the form purification transitions, including a refinement to strong X pffiffiffiffiffiffiffiffiffiffi purification transitions, unitary-dephasing channels, and ψ λ ⊗ ψ ⊗ l monitored channels that describe the models that have been j RS0E0 i¼ kpl jkRi j kli j i; ð10Þ l found to exhibit measurement-induced entanglement tran- k; sitions. The condition for a unitary-dephasing channel is where jk i, jψ li, and jli are all orthonormal states. By shown to be equivalent to a weaker version of the Knill- R k performing projective measurements of the operators Laflamme conditions for quantum error correction. In the most general formulation of the problem, we are X ψ ψ considering an open, quantum many-body system under- Ml ¼ j klih kljð11Þ going time evolution of the form k and applying the unitary operation depending on the N tðρÞ¼Tt ∘ T1ðρÞ; ð6Þ measurement outcome Uljψ kli¼jkSi, we can completely recover the initial state. The recovery operation acts only on where Ti are quantum channels or completely positive- S and can correct any initial state in support of ρ.Aswe trace-preserving (CPTP) maps [1]. The models we study discuss later, such arguments can be generalized to the below are random and Markovian in the sense that the Ti more physically relevant situation of approximate quantum are generated from independent, identically distributed error correction, where only ρRE0 approximately factor- ensembles, but we do not rely on this property in devel- izes [47,48]. oping the concept of a purification transition. To quantify the notion of recoverability or reversibility of As illustrated in Fig. 1(c), we can always consider a the system in more general settings, we introduce the N purification of a quantum channel by using a three- single-use quantum channel capacity defined in terms of component system consisting of a reference R, system S, the coherent quantum information, and an environment E. Within thisP framework, the initial mixed state of the system ρ λ k k is prepared 1 ¼ k kj Sih Sj Qð ÞðN Þ¼maxI ðρ ; N Þ; ð12Þ ρ c S through an entanglement operation between R and S, with S E in the computational zero state I ðρ ; N Þ¼Sðρ 0 Þ − Sðρ 0 Þ¼Sðρ 0 Þ − Sðρ 0 Þ; ð13Þ X pffiffiffiffiffi c S S RS RE E jψ i¼ λ jk i ⊗ jk i ⊗ j0i: ð7Þ RSE k R S where S ρ −Tr ρ log ρ is the von Neumann entropy k ð Þ¼ ½ [61], ρA is the reduced density matrix on A, and the ρ The open system dynamics is now modeled by an isometry maximum is taken over all density matrix ensembles S on SE UN called an isometric embedding, on the system S. An important identity that follows from ρ − ρ N ∶ ≥ 0 X pffiffiffiffiffi Eq. (13) is Sð SÞ Icð S; Þ¼IðR E0Þ , where IðR∶E0Þ is the mutual information between the reference jψ RS0E0 i¼ λkðIR ⊗ UN ÞjkRi ⊗ jkSi ⊗ j0i; ð8Þ k and the environment following the application of USE.For input states such that SðρSÞ¼IcðρS; N Þ, the mutual where N tðρÞ¼TrRE0 ½jψ RS0E0 ihψ RS0E0 j and the primes indi- information between the reference and the environment cate that UN has been applied to S and E. The original is exactly zero, which implies the existence of a perfect quantum channel dynamics is recovered by tracing over R recovery operation according to the previous argument. and E0. For simplicity, we focus on qubit models in which Thus, the single-use quantum channel capacity is the R, S=S0, and E=E0 consist of tensor-product Hilbert spaces maximum possible amount of quantum information that of two-level systems. can be perfectly transmitted with a single use of the noisy A crucial result in quantum information theory is that, channel. when the mutual information between the reference and the In the theory of quantum error correction, it is helpful to environment is zero, there exists a perfect recovery oper- generalize the single-use channel capacity to many copies ation acting only on the system that can recover the initial using the limiting definition entangled state jψ RSEi [46]. To review the argument, note 0 1 that zero mutual information between R and E implies that QðN Þ¼ lim Qð1ÞðN ⊗nÞ ≥ Qð1ÞðN Þ: ð14Þ their reduced density matrix factorizes n→∞ n 041020-6 DYNAMICAL PURIFICATION PHASE TRANSITION INDUCED … PHYS. REV. X 10, 041020 (2020) A finite value of Q implies that approximate quantum error mixed state averaged over measurement outcomes. To correction is possible on a space of asymptotic dimension strengthen the connections to our formal definitions, we 2nQ, even when it fails in a single use of the channel. present further numerical evidence in Sec. VI C that the However, this may come at the cost of performing state random Clifford model introduced in Sec. III A realizes a preparation (encoding) and decoding operations on the strong purification transition with parameter ac ≈ 1 for any n-fold replicated Hilbert space of the system [42–45].For pair of sequences ða; bÞ satisfying ac 041020-7 MICHAEL J. GULLANS and DAVID A. HUSE PHYS. REV. X 10, 041020 (2020) P † ¯ orthogonal density matrices ρm ¼ AmρAm=pm satisfying that there is a special set of error operators Ea ¼ b EbVba † that are uniquely identifiable, Tr½ρmρm0 ∝ δmm0 , where pm ¼ Tr½AmAmρ . As a result, the channel T always has at least one ensemble that can be i ¯ † ¯ λ δ δ unraveled by making projective measurements onto the hcijEaEbjcji¼ a ab ij: ð22Þ image of Am. We call a channel satisfying Eq. (17) a unitary-dephasing channel because it can be represented by If we interpret the Kraus operators in the quantum channel unitary dynamics followed by dephasing of some off- as possible error operators, then the unitary-dephasing diagonal coherences in the density matrix. condition in Eq. (17) essentially removes the constraint N from the Knill-Laflamme conditions that Eqs. (21) and (22) For composite channels t formed from unitary- δ dephasing channels, there is a natural realization of quantum are proportional to ij. This weaker condition implies that trajectories using what we call the monitored channels, the errors are detectable, even though the code words are not protected. In this setting, an intuitive picture for a strong purification transition in a unitary-dephasing channel is N ⃗ ðρÞ¼M ∘ T ∘ … ∘ M1 ∘ T1ðρÞ; ð18Þ m;t mt t that, in the mixed phase, the system spontaneously projects † the states of the system into code words that restore the full M ðρÞ¼P ρPm ⊗ jm ihm j; ð19Þ mi mi i i i Knill-Laflamme conditions [64]. As we show in Sec. IV,in general, this is achieved only approximately [65–68]. where Pmi is an isometric projector onto the image of Ami P † Monitored channels naturally arise in the weak meas- in a representation of T ðρÞ¼ A ρAm that satisfies i mi mi i urement picture for unitary-measurement dynamics that use N Eq. (17). The m;t⃗ are completely positive trace- ancilla qudits to perform the measurements of the system nonincreasing maps that act on the input state as (see Sec. V and also Refs. [29,30,41,69]). In this repre- sentation, the computational basis states of the ancilla are N ρ ρ † ⊗ ⃗ ⃗ m;t⃗ ð Þ¼Km⃗ Km⃗ jmihmj; ð20Þ entangled with all possible paths in a quantum trajectory evolution and then “dephased” by an environment. Thus, for Km⃗ ¼ Amt Am1, thereby directly realizing a quantum the quantum trajectory can be unraveled by an observer trajectory. We refer to the (trace-preserving)P complete through projective measurements of the ancillae in their N u N mixture of monitored channels t ¼ m⃗ m;t⃗ as an computational basis. Several special properties of this class unraveled channel. We define a set of monitored channels of channels were exploited in recent works on the meas- as having a (strong) purification transition if its unraveled urement-induced entanglement transitions [30,41,69]. channel has a (strong) purification transition in its quantum Here, we show that imposing the strong purification capacity or channel-averaged quantum capacity. Note that condition on a monitored channel is enough to show that unraveled channels have the special property that they are the monitored dynamics gives rise to an optimal encoding unitary-dephasing channels for every choice of t. operation for the future evolution of the channel. Unitary-dephasing channels are examples of “degrad- More specifically, we consider a monitored channel ” N ∘ ∘ ∘ ∘ able quantum channels, which are defined by the con- m;t⃗ ¼ Mmt Tt M1 T1 that exhibits a strong ρ dition that E can be obtained by a CPTP map acting on the purification transition. We prove that N m;t⃗ defines a family system. As shown by Devetak and Shor, the quantum of quantum-error-correcting codes that can be efficiently N channel capacity of a degradable channel is equal to the generated with a single use of m;t⃗ a for any allowed single-use channel capacity QðN Þ¼Qð1ÞðN Þ [44]. This sequence of pairs ða; bÞ in the thermodynamic limit. These identity helps simplify the calculation of Q, but it does not encodings have an associated high-fidelity, single-use provide explicit constructions for the encoding and decod- recovery operationP for the future evolution of the unraveled u ing operations or imply that they can be done with a single channel N ¼ ⃗M ∘ T ∘ ∘ M ∘ T 1 on tb;ta m mtb tb mtaþ1 taþ copy of the system [63]. the polynomial timescale tb − ta in the thermodynamic Before continuing, we discuss an interesting connection limit. The amount of encoded information is optimal in the between the unitary-dephasing condition in Eq. (17) and a u sense that it converges to the channel capacity of N t for foundational result in quantum error correction known as any ta ≤ t ≤ tb. Our proof is based on randomly sampling the Knill-Laflamme conditions [1,51]. Given a set of pure codes generated by the channel dynamics that have the quantum states, or code words, jcii, and a set of error desired properties on average, which implies the existence operators Ea, the necessary and sufficient conditions for of a large subset of the codes with the desired properties. these errors to be correctable are † IV. DYNAMICALLY GENERATED CODES: hc jE E jc i¼C δ ; ð21Þ i a b j ab ij EMERGENT QUANTUM ERROR CORRECTION where Cab is an arbitrary Hermitian matrix. Changing into In this section, we describe in greater detail the proper- † λ δ an orthonormal basis for C, ðV CVÞab ¼ a ab, implies ties of the quantum-error-correcting codes in the mixed 041020-8 DYNAMICAL PURIFICATION PHASE TRANSITION INDUCED … PHYS. REV. X 10, 041020 (2020) phase in both many-copy and single-use quantum-error- Thus, within the encoded subspace, the dynamics of the correction protocols. Our discussion of many-copy error channel acts as a type of effective, reversible unitary correction follows directly from well-known existence dynamics. proofs of encoding and decoding pairs that can achieve In the thermodynamic limit, we do not expect to require the channel capacity of any quantum channel in the the infinitely replicated Hilbert space to construct high- asymptotic limit of an infinite number of copies [42,45]. fidelity encoding and decoding operations, provided one The results of single-use quantum error correction are replaces the strong reversal condition in Eq. (23) by a specific to the class of unitary-dephasing channels intro- weaker, probabilistic reversal condition [47,68]. In this duced in the previous section and, to our knowledge, have case, it is likely that any system exhibiting a strong not previously been discussed in the literature. purification transition can be encoded or decoded on the As we noted above, a key feature of a purification allowed polynomial timescales; however, we have not transition is the associated existence of a quantum-error- found a rigorous proof of this result. Instead, we establish correcting code on the replicated Hilbert space H⊗n, which a version of this result for the special case of monitored can be decoded on all polynomial timescales. To make this channels N m;t⃗ with a strong purification transition. statement more precise, we let N t ¼ Tt ∘ ∘ T1 be a Monitored channels include all models known to exhibit family of quantum channels indexed by integers t with a measurement-induced entanglement transitions; however, purification transition. We fix ϵ > 0 and b with 1 041020-9 MICHAEL J. GULLANS and DAVID A. HUSE PHYS. REV. X 10, 041020 (2020) quantum-error-correcting code for the future evolution of to this case with only minor modifications. In cases where the channel. We remark that this result was recently the channel capacity density is zero, such as in the pure anticipated in Ref. [49] (without making connections to phase ðp>pcÞ, Theorem 1 still applies; however, when channel capacities), where it was referred to as a “self- the channel capacity is not just subextensive but strictly organized” quantum error correction. Similar to the exist- zero (or very close), the proof of the theorem applies for the ence of the single-shot encoding-decoding pair discussed trivial reason that there is no code space where information above, this optimal code-generation process may generi- can be stored and the recovery operation only needs to cally apply to systems exhibiting a strong purification approximately succeed on a single-input state. transition; however, we are unaware of a rigorous proof for When the monitored channels N m;t⃗ are drawn from a this conjecture. Such a result would have broader impli- random ensemble (i.e., random channels), as discussed in cations for fault-tolerant quantum computing because, Sec. III C, we define a channel-averaged strong purification intuitively, it would suggest that the dynamics of a fault- transition with respect to a fixed input state ρ0 by imposing tolerant system may generically project the system into an the same convergence conditions on the channel-averaged ¯ optimal code space for its future evolution. We leave a more quantum capacity Imax. Since our proof is based on a complete study of these possibilities for future work. We Markov inequality, the theorem directly generalizes to the summarize our results on emergent quantum error correc- channel- and code-averaged code rate and recovery fidelity. tion in monitored channels in the following theorem: In models such as the random Clifford model, based on N ∘ ∘ ∘ ∘ Theorem 1: (Informal) Let m;t⃗ ¼Mmt Tt Mm1 T1 random unitary circuits interspersed with projective mea- be a monitored channel indexed by measurement outcomes surements [25–27], it is convenient to study channel- m⃗and integers t>0 with a strong purification transition. averaged strong purification transitions for the completely ρ N There is an input state 0 such that the density matrices mixed input state ρ0 ¼ I=2 . The completely mixed state is ρ ∝ N ρ m⃗ m;t⃗ a ð 0Þ obtained from evolving the monitored a natural input state for these models because its channel- dynamics define optimal (capacity-achieving) quantum- averaged coherent quantum information often saturates ¯ error-correcting codes forP the future evolution of the Imax (see Sec. III C). N u N ρ → A ρ unraveled channel t;ta ¼ m⃗ m;t;t⃗ a , i.e., Sð m⃗Þ=N We note that the entanglement fidelity Feð ; Þ¼ (I ⊗ A Ψ Ψ Ψ ) cðpÞ, and there is a high-fidelity reversal operation Rmt⃗ a;t F R ðj RSih RSjÞ; j RSi is defined as the mixed- ∘N u ρ →1 Ψ ρ such that the entanglement fidelity FeðRmt⃗a;t t;ta ; m⃗Þ state fidelity between a purification j SRi of and its image A in the thermodynamic limit for ta ≤ t ≤ tb. Here, cðpÞ is under the channel [77]. It is independent of the specific Ψ the channelP capacity density of the unraveled channel choice of j SRi and quantifies the degree of quantum u N ¼ ⃗N ⃗ . coherence of the channel. For example,P for any ensemble t m m;t λ ρ λ The formal statement and proof of the theorem are given representation e ¼f k; jkig of ¼ k kjkihkj, Fe is a in Appendix A. This result is intuitively expected, but the lower boundP on the average fidelity for state transmission ¯ A λ A ≥ A ρ proof also provides useful bounds on the rate of conver- Fð ;eÞ¼ k kF½ ðjkihkjÞ; jki Feð ; Þ. As a result, gence of the entanglement fidelity. We show in Appendix A a direct consequence of our theorem is that randomly ρ how to use these bounds, in combination with our numeri- chosen input states from an arbitrary ensemble for m can cal results in Sec. VI C, to determine when the random be evolved both forward and backward in time on all Clifford model can be approximated by a random unitary polynomial timescales in the thermodynamic limit [78]. circuit for p 041020-10 DYNAMICAL PURIFICATION PHASE TRANSITION INDUCED … PHYS. REV. X 10, 041020 (2020) ρ According to Theorem 1, each Ataþ1ljm can be approxi- A natural unraveling into quantum trajectories fpm⃗; m⃗g mated in the thermodynamic limit by an isometry into a set for the channel dynamics takes the form of orthogonal spaces indexed by ljm. Let ϵ > 0 and N be X † sufficiently large that the average recovery fidelity is ρ ⃗ ¼ K ⃗ρK =p ⃗; N ðρÞ¼ p ⃗ρ ⃗; ð31Þ ¯ m m m⃗ m t m m Fe ≥ 1 − ϵ=ðtb − taÞ. By immediately performing a pro- m⃗ ρ jective measurement on the system into the support of ljm, † ρ we can approximately map the dynamics to one of these where pm⃗ ¼ Tr½Km⃗Km⃗ is the probability of a given isometries up to an average error less than ϵ=ðtb − taÞ [79], trajectory. An equivalent description of the trajectory is, in terms of a quantum circuit, interspersed with measure- ∘ ψ ψ → ψ Mljm Ttaþ1ðj kmih kmjÞ Utaþ1ljmj kmi; ð28Þ ments, as depicted in Fig. 3. We show in Appendix B that the coherent quantum ψ ρ for j kmi sampled from any ensemble for m.Byan information of the full channel N t is bounded by the inductive argument, we can perform a similar mapping average entropy of any trajectory ensemble, for each subsequent channel Tt up until t ¼ tb while X keeping the total average error bounded by ϵ. I ðρ; N Þ ≤ p ⃗Sðρ ⃗Þ: ð32Þ N c t m m As a result, the evolution of the channel t interspersed m⃗ with projective measurements induces time-local unitary dynamics on the code space in the thermodynamic limit, This bound can be used to constrain whether the underlying with the usual group structure of unitary evolution. channel N is in a mixed or pure phase. To show this ρ t However, the code-space density matrix m is continually constraint, we need to consider the coherent quantum evolving in a manner that depends on the measurement ⊗n information of the replicated channel N t . Applying the l l … l ⊗ outcomes t ¼ð taþ1; ; tÞ. Consequently, the unitary n bound in Eq. (32) to N t and using subadditivity of gates in each time step can depend on the full history of the entropy results in the bound previous gates applied to the system, which is consistent X X with the more general history dependence of Eq. (27).We ⊗n ðiÞ Icðρ; N t Þ ≤ p m⃗ Sðρ ⃗ Þ; discuss further implications of these results in Sec. VIII B. f jg fmjg i fm⃗jg ðiÞ −1 † ρ ¼ p ⃗ Trl≠ ½K ⃗ ρK ; V. MONITORED RANDOM CIRCUITS fm⃗jg fmjg i fmjg fm⃗jg ⊗ ⊗ In this section, we discuss monitored channels defined in Kfm⃗jg ¼ Km⃗1 Km⃗n ; ð33Þ terms of random quantum circuits. We also make more concrete connections between the monitored channel where ρ acts on the n-fold replicated Hilbert space dynamics and the underlying dynamics of the composite ðiÞ and ρ ⃗ is the reduced density matrix of the ith replica unitary-dephasing channel. fmjg We consider random quantum channels of the form conditioned on the measurement record across all n ⃗ X replicas fmjg. N ρ ρ † tð Þ¼ Km⃗ Km⃗; ð29Þ m⃗ System Unitary dynamics mt m1 Km⃗ ¼ UtPt U1P1 ; ð30Þ mi Measurement where Pi is a sequences ofP positive-operator-valued m I ⃗ measures (POVMs) that satisfy m Pi ¼ , m indexes the measurement outcomes, and Ui are unitary operators that can depend on the most recent measurement outcomes. Reference ∈ 0 1 We will focus on the case where mi f ; g takes one of Time m two possible outcomes and Pi are simple single-site projectors. These random channels have a decomposition Space into unitary-dephasing channels as defined in Sec. III C; however, whether the channel has a purification transition FIG. 3. Unitary-measurement dynamics in monitored random circuits. The system is initially partially entangled with a set of depends on the details of the unitary gates and the space- reference qubits and undergoes unitary dynamics interspersed time position of the projectors. In this work, we consider with measurements. The two phases in a purification transition the case where the projectors and unitaries all act on a fixed correspond to whether or not the measurements remove all number of qubits. Absent monitoring, such a channel will coherent quantum information of the input state on polynomial tend to drive the system towards infinite temperature. timescales in the thermodynamic limit. 041020-11 MICHAEL J. GULLANS and DAVID A. HUSE PHYS. REV. X 10, 041020 (2020) Now, consider the case where all initial states converge to unraveled channels, the inequality in Eq. (32) is satu- zero entropy density averaged along trajectories. If we fix rated [80], the measurement records m⃗l on replicas l ≠ i, then the X ðiÞ ρ N u ρ density matrix ρ can be interpreted as a single-copy Icð ; t Þ¼ pm⃗Sð m⃗Þ: ð36Þ fm⃗jg m⃗ quantum trajectory that occurs with a conditional proba- ⃗ ⃗ l ≠ bility pðmijml; iÞ. Averaging over these conditional We showed in Sec. III A that the random Clifford model is a Pprobabilities implies that the average entropy density of monitored channel with a purification transition. We show ðiÞ ⃗ pðm⃗jm⃗l; l ≠ iÞSðρ Þ must also converge to zero below that it also satisfies the conditions needed for a mi i fm⃗jg for each value of the other measurement records. As a strong purification transition. Applying Theorem 1 to this result, each term in the upper bound in Eq. (33) converges model then shows that, for p 041020-12 DYNAMICAL PURIFICATION PHASE TRANSITION INDUCED … PHYS. REV. X 10, 041020 (2020) A. Entanglement transition I3ðA∶B∶CÞ¼IðA∶BÞþIðA∶CÞ − IðA∶BCÞ. For pure ∶ ∶ One of the central findings of our numerical study of this states, I3ðA B CÞ is symmetric under all permutations purification transition in 1 þ 1 dimensions is that it occurs of ðA; B; C; DÞ, where D is the rest of the sample. The concurrently, and with the same critical exponents, as the tripartite mutual information, sometimes referred to as the entanglement phase transition for pure initial states. Thus, topological entanglement entropy [31,32], is a natural before examining the scaling behavior of the mixed-state measure of the degree to which information in a quantum 8 dynamics, we first revisit the critical properties of the wave function is encoded nonlocally. Similar to the L= entanglement phase transition for pure initial states. antipodal mutual information, it has the effect of removing In Ref. [28], it was shown that the critical region of the the logarithmic divergences that appear in the half-cut entanglement phase transition can be precisely identified by entanglement at the critical point, which reduces finite-size corrections to scaling. We find that hI3ðA∶B∶CÞi vanishes looking at the mutual information IðA∶BÞ¼SðρAÞþ Sðρ Þ − Sðρ ∪ Þ between two antipodal regions on the in the pure phase, approaches a universal constant of about B A B −0 5 circle of length L=8. We have found that a similar, but . at the critical point, and has the expected negative more accurate, probe of the critical point is to use the volume-law scaling in the mixed phase. This behavior can tripartite mutual information between three contiguous be explained using a minimal cut picture for the entangle- regions of length L=4 [see inset to Fig. 4(a)], defined as ment scaling near the critical point [82]. In Fig. 4(a),we show the finite-size scaling near the critical point for I3 (a) -1 -10-2 -10 starting from pure initial states. We observe a clear crossing of I3 vs p with increasing L, which we use to identify pc ¼ 0.1593ð5Þ. From collapsing the L ¼ 128–512 data with this value of p , we extract ν 1.28 2 . These -1 c ¼ ð Þ -10 -100 estimates are consistent with those reported in Ref. [28]. Another basic quantity of interest in characterizing the 0.15 0.16 −ν transition is the correlation length ξ ∝ jp − pcj .To -100 realize a quantitative measure of the correlation length L = 64 I A∶B L = 128 on both sides of the transition, we study ð Þ for the L = 256 region shown in an inset of Fig. 4(b), where A and B are of L = 512 equal size, r 041020-13 MICHAEL J. GULLANS and DAVID A. HUSE PHYS. REV. X 10, 041020 (2020) (a) values of the scaled time. In all cases, we see an excellent 2 10 collapse of the data for L ranging from 64 to 512. C. Structure of optimal code-space density matrices 0 ) 10 As shown in Theorem 1 in Sec. IV, the late-time density matrix defines an optimal quantum-error-correcting code S( space for the channel dynamics, which motivates the study of the structure of its correlations. In addition, these studies 10-2 help make a more direct comparison between the purifi- cation phase transition for mixed initial states and the -3 -2 -1 0 1 entanglement phase transition seen for pure initial states. 10 10 10 10 10 A convenient diagnostic of the mixed-state density matrix is the bipartite mutual information IðA∶AcÞ for A 2 ∶ c (b) 10 of varying length r. For pure states, IðA A Þ reduces to twice the entanglement entropy. The main qualitative features are illustrated in Fig. 6(a).Forp ≥ pc, the mixed state purifies on a timescale at most linear in L, which implies that pure and mixed initial conditions have identical 0 c ) 10 late-time scaling behavior for IðA∶A Þ. In the pure or area- S( law phase, the long-time states exhibit only area-law mutual information, which quickly converges to a constant value independent of L. At the critical point, the system builds up logarithmic mutual information. The most sig- 10-2 nificant result presented here is that, for p 041020-14 DYNAMICAL PURIFICATION PHASE TRANSITION INDUCED … PHYS. REV. X 10, 041020 (2020) thermodynamic limit. These two conditions appear specific to the optimal code-generation process because starting with an entropy density significantly below the channel capacity limit leads to a much more rapid convergence (not shown) to the plateau value. For these input states that do not maximize the channel-averaged quantum capacity, the likely strong purifi- cation parameter is ac ≈ 0 for any pair of sequences ða; bÞ with 0 041020-15 MICHAEL J. GULLANS and DAVID A. HUSE PHYS. REV. X 10, 041020 (2020) 1 þ 1 dimensions, as argued by Fan et al., but not necessarily for higher dimensions as we observe an explicit violation in an all-to-all model. A key property of a degenerate code is that it can still have an error-correction threshold even when the distance does not scale extensively in the system size. We can sensitively test this aspect of the codes in the volume-law phase because we can generate the optimal ensemble of codes with respect to the code rate. By computing a quantity we call the contiguous code length, we show in Appendix F that the average distance of the optimal codes in 1 þ 1 dimensions is subextensive for p 041020-16 DYNAMICAL PURIFICATION PHASE TRANSITION INDUCED … PHYS. REV. X 10, 041020 (2020) The qualitative behavior of hSQi in the mixed phase is close proximity to a transition from Poisson to Wigner- shown in Fig. 7 for p ¼ 0.30. To set up these simulations, we Dyson level statistics in the entanglement spectrum for pure choose an initial state given by a random Clifford state; we states [89]. For stabilizer circuit models, the Hartley entropy then measure one site in the system and place this qubit in a is not a good diagnostic for pcc because the entanglement maximally entangled state with the reference qubit. Similar spectrum is always degenerate. to the purification dynamics of the completely mixed state, For Haar-random circuits with brickwork geometry in the reference qubit purifies for a short period that is 1 þ 1 dimensions, the entanglement transition for the independent of N before crossing over to a late-time plateau. von Neumann entropy occurs at a much lower value The persistence time of this plateau diverges exponentially (pce ≈ 0.17) than the analytical result for 2D percolation in N over this range of sizes. The results of these numerics pcc ¼ 1=2 [26,82]. Interestingly, Bao, Choi, and Altman indicate that 0.30 041020-17 MICHAEL J. GULLANS and DAVID A. HUSE PHYS. REV. X 10, 041020 (2020) purification transitions in 1 þ 1 dimensions have an emer- measurements, whereas in our analysis, we have taken all gent conformal symmetry [28,41,69,83]. The appropriate operations to be implemented in an ideal manner that is classification of these conformal field theories (CFTs) is not perfectly known to the observer. Far away from pc, where currently known. Although certain limiting cases are equiv- the correlation length ξ is much less than the system size, alent to percolation [26,41,69], as noted in Ref. [28], the conventional scaling arguments suggest that the encoding observed value of αðpcÞ for stabilizer circuits differs from and decoding operations away from the channel capacity that of 2D critical percolation by roughly a factor of 3, limit will already converge to high fidelity after depth suggesting that these models might be in different univer- t=ξz ∼ log N in the thermodynamic limit, where z is the sality classes. More recently, Li, Chen, Ludwig, and Fisher dynamical critical exponent (z ¼ 1 for the 1 þ 1-dimen- have found a variety of other surface scaling dimensions for sional model studied above). Although the log N scaling the “Clifford” CFT that differ from percolation [83].Our implies that the encoding is unlikely to be directly estimated bulk correlation length exponent ν ¼ 1.28ð2Þ,on achievable in a fault-tolerant manner, such low-depth codes the other hand, is very close to the ν ¼ 4=3 of 2D with high code rates may be crucial in improving the percolation. One possible explanation for the small differ- performance and flexibility of intermediate-scale quantum ence in ν is that the Clifford critical point can be obtained by devices [91]. Moreover, simple extensions of the present a weak perturbation of a percolation fixed point [69]. models to include active feedback may allow the realization We have also introduced a method to determine the order of fully fault-tolerant encoding schemes in this framework. parameter exponent for the purification transition in these We still have the problem of efficiently characterizing the models [40]. Bulk order parameter exponents in 1 þ 1 logical operator algebra of the code-space density matrices ρ R dimensions appear to be consistent with percolation for m or implementing the recovery operations mta;t in an both stabilizer circuits and more general models [40,82], efficient manner. Characterizing and finding recovery but there is some evidence for a small, but significant, operations for approximate error correction of quantum difference in the surface order parameter exponent in both channels is an active area of research in quantum informa- cases [40,82,83]. It has been argued that these critical tion theory (e.g., see Refs. [92,93]). Although our proof is theories are described by logarithmic conformal field nonconstructive for the recovery operation, the bound that theories with central charge zero [69]. Such theories have we prove on its entanglement fidelity implies that there are logarithmic corrections to scaling [90], so we should be explicit recovery maps expressed in terms of ρm and the N cautious in interpreting small differences in apparent t;ta that also satisfy the conditions of Theorem 1 [47]. critical exponents. Furthermore, in the case of stabilizer circuits exhibiting a Many questions remain about the characterization of strong purification transition, the logical operators and these phase diagrams and universality classes, particularly recovery operations can be efficiently computed using outside 1 þ 1 dimensions or in the presence of quenched the Gottesman-Knill theorem [52,53]. Because of the large disorder. The ubiquity of these phases in open quantum number of advantages of stabilizer codes for FTQC, the system dynamics motivates the development of a deeper codes we have found in stabilizer circuits are likely the understanding of the phase diagrams, the defining properties most promising to explore in this context. of each phase, and the critical points. The close connections Our deconstruction of the recovery operation reduces the that arise between these problems and fault-tolerant quan- decoding problem in the mixed phase to the problem of tum computation are also interesting to explore. In the near characterizing the quantum circuits U l U l aris- t2 t2 jm t1 t1 jm term, these efforts should aid in characterizing the physics of ing from the mappings of the quantum channel dynamics noisy intermediate-scale quantum devices. ∘ → Mlijli−1m Ti Uilijm. Provided one can efficiently com- pute or approximate U l for each combination ðt; l ;mÞ, B. Applications to fault tolerance t tjm t the full evolution in the mixed phase can be described by a A significant consequence of Theorem 1 on dynamically known unitary circuit acting on the system. In most cases, we generated codes is that, under the stated conditions, it expect that the correlations between measurement outcomes provides a rigorous and efficient method to sample will be short ranged for p 041020-18 DYNAMICAL PURIFICATION PHASE TRANSITION INDUCED … PHYS. REV. X 10, 041020 (2020) of the mixed phase may provide useful insight into decoding The existence of highly efficient classical algorithms for problems for standard quantum error correction (and these dynamics is crucial in benchmarking the performance vice versa). of the experimental device in a scalable manner. Although We have argued that there is substantial motivation to one might argue that the experiment does not probe new better characterize the class of stabilizer and nonstabilizer physics because its ideal dynamics can be classically codes that arise in the mixed phase. Studying the associated simulated, this is far from the case. The actual physical recovery operations for the channel dynamics may also evolution of the experimental system differs dramatically provide insight into decoding for quantum error correction from the simplistic simulations used for stabilizer circuits. in more conventional scenarios. Because of their local Stabilizer states are a rich class of quantum wavefunctions encoding schemes, degeneracy, and optimal trade-offs that can have extensive entanglement and realize many of between high code rates and error thresholds, these codes the intrinsic subtleties associated with quantum mechanical have potential advantages for fault-tolerant quantum com- systems that do not have natural classical analogs. putation with low overhead [91]. Furthermore, there are many theoretical reasons to suspect that fully fault-tolerant quantum computing will rely, at C. Experiments in near-term quantum devices some level, on stabilizer-based quantum-error-correcting Current quantum computing devices are far from codes. By studying the statistical physics of stabilizer meeting the full requirements for scalable fault-tolerant dynamics, one is arguably gaining fundamental insight operation [94]. The advantage of studying this class of into the phase of matter associated with a fully fault- unitary-measurement dynamics in such near-term, inter- tolerant quantum computer [96]. mediate-scale quantum (NISQ) devices is that it combines most of the necessary ingredients to realize general-purpose IX. CONCLUSIONS quantum error correction but in a stochastic, unstructured We have demonstrated the existence of a class of setting. Similar to the recent experiments on “quantum dynamical purification phase transitions between one phase supremacy” in random circuit sampling [95], we suspect where the many-body dynamics purifies arbitrary initial that realizing scalable demonstrations of measurement- states at a rate independent of the system size, and another induced entanglement transitions would push towards phase where the time to purify grows exponentially in the achieving what one might call “fault-tolerant quantum size of the system. We proved several general results on this supremacy.” Such intermediate milestones are crucial in class of phase transitions. To gain deeper insight into these the drive towards scalable quantum computing. problems, we studied specific models of stabilizer circuits In this vein, the purification dynamics introduced in this that are amenable to large-scale numerics; however, the work serves as a useful probe of these phase transitions general features underlying the phase transition are relevant outside a fully fault-tolerant setting. The exact realization for many physical systems undergoing high-fidelity, con- of the ordered phase in d spatial dimensions requires a limit tinuous monitoring, interspersed with entangling dynamics. 1 where the local decoherence rate γ ≪ 1=Ldþ or smaller. In Our observation that the bipartite mutual information for contrast, assuming the local entropy production rate density completely mixed initial states grows sublinearly in time scales as γ, a crossover between the mixed and pure phases may aid in the development of efficient classical decoders should persist at late times with a crossover length scale ξc for 1D nonstabilizer models using matrix product methods. that naively scales as γ−1=ðdþ1Þ for low enough d.In Furthermore, such purification transitions naturally arise addition, the purification dynamics displays a direct sig- in systems with long-range interactions, which are relevant nature of the two phases already at the level of a single to quantum computing platforms based on trapped ions reference qubit [40], which, together with the quantum [97] or quantum networks [98]. The perspective on the Fisher information [41], avoids the extensive overhead in measurement-induced entanglement transitions in terms of both experiment and data analysis associated with meas- purification dynamics may also aid in experimental studies, uring entanglement or entropy of large regions. Using these where imperfections almost inevitably drive the system local probes, the mixed and pure phases can be efficiently towards mixed states. studied in general models away from pc with constant Finally, together with Ref. [30], we have shown that the depth circuits t ≫ ξ and a large enough number of qubits to existence of the volume-law or mixed phase, where the suppress finite-size effects [40]. Such improvements in the entropy density remains nonzero, implies an extensive error resilience of quantum algorithms can be the difference quantum channel capacity in the underlying open system between applications on near-term devices and being dynamics. In this work, we further established that the pushed into the future realm of full fault tolerance. monitored dynamics itself can be used to optimally encode Although not strictly necessary [40], the most natural quantum information against the future nonunitary evolu- setting to begin studying these dynamics in near-term tion of the system. These results have broad implications quantum computing devices is by implementing stabilizer for the study of measurement-induced phase transitions. circuit models such as the ones considered in this work. In particular, they imply that there is a “code space” of 041020-19 MICHAEL J. GULLANS and DAVID A. HUSE PHYS. REV. X 10, 041020 (2020) operators in the system that are not measured and, thus, can N ρ ∘ ∘ ∘ ∘ ρ m;t⃗ ð Þ¼Mmt Tt Mm1 T1ð Þ; ðA1Þ preserve quantum information about the initial state. As a result, the time evolution in the ordered phase at late times † M ðρÞ¼P ρPm ⊗ jm ihm j: ðA2Þ becomes effectively reversible unitary dynamics in the mi mi i i i thermodynamic limit. The dynamically generated codes The formal statement of the theorem is as follows: are of intrinsic interest in quantum information because Theorem 1. Let N ⃗ ¼ M ∘ T ∘ ∘ M ∘ T1 be they saturate fundamental bounds on the trade-off between m;t mt t m1 a monitored channel indexed by measurement outcomes m⃗ the density of encoded information and the threshold error and integers t>0 with a strong purification transition. For rate, with potential applications to fault tolerance. It would any p 041020-20 DYNAMICAL PURIFICATION PHASE TRANSITION INDUCED … PHYS. REV. X 10, 041020 (2020) ρ N ρ We now show how this bound can be used to construct a Taking m ¼ m;ta ð 0Þ=pm, from Eq. (A5) and our choice of input state, we can establish the bound on the high-fidelity recovery channel following Ref. [48].For average code rate at time ta, completeness,P we reproduce the full argument. First, we let Ψ λ k ϕ ρ j SRjmi¼ k kjmj Rjmij kjmi be a purification m into X orthonormal states. Recall that F ρ; σ is defined as the pm ð Þ Sðρ Þ − cðpÞ ρ σ N m maximum square overlap over all purifications of and . m ρ We fix a purification of REjm given by the state obtained u ¼jI ðρ0; N Þ=N − cðpÞj Ψ c ta from the evolution of j SRjmi under the isometric embed- ≤ − − ϵ ding UN , maxðjQta =N cðpÞj; jQtb =N cðpÞjÞ < ; ðA8Þ t;ta X ffiffiffiffiffiffiffiffiffiffiffi Ψ p ϕ l where the second line follows from the fact that Ic is j RSEjmi¼ pkljm jkRjmij kljmij jmi; ðA14Þ monotonically decreasing under quantum operations (e.g., kl see Ref. [46]). where jljmi is the state of the environment conditioned on Now, considerP p theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi continued dynamics of the purifica- ψ ψ ρ observing state jmi at time ta. Note that this case is a tion j mi¼ k pkm=pmjkRij kmi of m under an iso- N u different representation of the evolved state from Eq. (A9). metric embedding of t;ta , ˆ By definition, there exists some purification jΨ i of rffiffiffiffiffiffiffiffiffiffi RSEjm X ρ ⊗ ρ that saturates the fidelity pkml Rjm Ejm UN u jψ ij0i¼ jk ijψ lijli; ðA9Þ t;ta m p R km kl m ρ ρ ⊗ ρ Ψˆ Ψ 2 Fð RE; R EÞ¼jh RSEjmj RSEjmij : ðA15Þ where l ∈ fðm 1; …;mÞg. The mutual information taþ t Ψˆ between the reference system and the environment for this The state j RSEjmi will have a Schmidt decomposition of future evolution is given by the form X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∶ ρ − ρ N u Ψˆ λ ϕˆ l IðR EjmÞ¼Sð mÞ Icð m; t;ta Þ j RSEjmi¼ kjmpljm jkRjmij kljmij jmi; ðA16Þ X l ρ − ρ k ¼ Sð mÞ pljmSð mlÞ; ðA10Þ l with all states in the threefold tensor product orthonormal. This purified state can be corrected back to the original where pljm ¼ pml=pm is the conditional Born probability entangled state jΨ i by a projective measurement onto l SRjm P of measurement record conditionedP on the prior meas- ρ ϕˆ ϕˆ the support of ljm, e.g., Pljm ¼ k j kljmih kljmj, fol- urement record m and pml ¼ k pkml. The key step in the proof is that the average mutual information is then given lowed by an isometry X by the decrease in coherent quantum information † ˆ U ¼ jϕ ihϕ l j; ðA17Þ X tljm kjm k jm k IðR∶EjmÞ¼ pmIðR∶EjmÞ m which was also used in Eq. (28) of Sec. IV. Applying this ρ N u − ρ N u ϵ2 ¼ Icð 0; ta Þ Icð 0; t Þ < ; ðA11Þ recovery operation, which acts only on S, to the reduced ρ Ψ Ψ density matrix RSjm ¼ TrE½j RSEjmih RSEjmj results in a where the bound follows because Qta;b is an upper or lower corrected state of the system and reference u bound on Icðρ0; N t Þ for all ta ≤ t ≤ tb and we take N X Q − Q < ϵ2 R ρ † ρ ρc sufficiently large that j ta tb j . mta;tð RSjmÞ¼ UtljmPljm RSjmPljmUtljm ¼ RSjm: Since mutual information is non-negative, we can use a l Markov inequality to bound the probability that it is larger than ϵ, Since the fidelity is monotonically increasing under quan- tum operations [70], this result implies the bound on the entanglement fidelity P½IðR∶EjmÞ > ϵ ≤ IðR∶EjmÞ=ϵ < ϵ: ðA12Þ F ðR ∘ N ; ρ Þ¼Fðρc ; jΨ iÞ When IðR∶EjmÞ < ϵ, this implies that the reduced density e mta;t t;ta m RSjm RSjm ≥ Ψˆ Ψ matrix of the reference and environment is close to a Fðj RSEjmi; j RSEjmiÞ product state in the sense that ρ ρ ⊗ ρ ¼ Fð REjm; Rjm EjmÞ pffiffiffi pffiffiffi ρ ρ ⊗ ρ ≥ 1 − 2 ϵ ≥ 1 − 2 ϵ Fð REjm; Rjm EjmÞ : ðA13Þ : ðA18Þ 041020-21 MICHAEL J. GULLANS and DAVID A. HUSE PHYS. REV. X 10, 041020 (2020) X pffiffiffiffiffiffiffiffiffiffiffiffi Using Eqs. (A12) and (A18), we can bound the average jψ RS0E0 i¼ λkpkm⃗jkRi ⊗ jψ km⃗i ⊗ jm⃗i; ðB2Þ entanglement fidelity as stated in the theorem. □ m;k⃗ The theorem is nonconstructive for ρ0 and R mta;t pffiffiffiffiffiffiffiffi (because of the assumption of channel capacity saturation Km⃗jkSi¼ pkm⃗jψ km⃗i: ðB3Þ ˆ of ρ0 and the reliance on jϕkmli, respectively) but otherwise explicit. The theorem can be generalized to the case where The coherent quantum information satisfies the identity ρ0 does not saturate the channel capacity, but it has a strong ρ − ρ N ρ ρ − ρ purification transition with respect to ρ0 [i.e., the coherent Sð SÞ Icð S; tÞ¼Sð RÞþSð E0 Þ Sð RE0 Þ quantum information I ρ0; N converges to an extensive 0 cð tÞ ¼ IðR∶E Þ¼DðρRE0 jρR ⊗ ρE0 Þ; value with time-independent subextensive corrections]. This generalization is useful in numerical studies or experi- where primes denote the reduced density matrix after ments where capacity-achieving ensembles can be difficult applying I ⊗ USE (note, ρR0 ¼ ρR), IðA∶BÞ is the mutual to find or initialize. As discussed in Sec. VIII, the existence information between subsystems A and B, and DðρjσÞ¼ of this recovery operation implies that there is also an −Tr½ρðlog σ − log ρÞ is the relative entropy. The relative ρ explicit recovery operation expressed in terms of m and entropy is monotonically decreasing under the action of N u t;ta with similar fidelities [47]. quantum channels due to strong subadditivity of quantum In Sec. IV, we introduced a stronger condition that the entropy [76]. Applying the dephasing channel to the recovery map from tb to ta can be approximately decom- environment, posed into a time-local sequence of isometries acting on the X system. To avoid a large buildup in the error, we require that EðρEÞ¼ hm⃗jρEjm⃗ijm⃗ihm⃗j; ðB4Þ the average entanglement fidelity for the recovery operation m⃗ ¯ at each time step satisfies Fe > 1 − ϵ=ðtb − taÞ. We can see how to satisfy this condition from the proof of Theorem 1. results in the identity In particular, in Eq. (A11), we can impose the condition ρ − ρ N − ϵ4 16 − 4 Sð SÞ Icð S; tÞ that jQta Qtb j < = jtb taj , which ensures that ¯ Fe > 1 − ϵ=ðtb − taÞ. As a result, this decomposition is ¼ DðρRE0 jρR ⊗ ρE0 ÞðB5Þ − 1 − satisfied when jQta Qtb j decays to zero faster than =jtb 4 ≥ I ⊗ E ρ ρ ⊗ E ρ taj in the thermodynamic limit. For the strong purification D½ð Þð RE0 Þj R ð E0 Þ ðB6Þ transition in the random Clifford model, we found in X Sec. VI C that the difference in channel-averaged qua- ¼ SðρRÞ − pm⃗Sðρm⃗Þ: ðB7Þ ≫ ¯ − ⃗ ntum capacity scales at late times t L as jImaxðtaÞ m ¯ ∼ ≈ − ImaxðtbÞj lnðtb=taÞ ðtb taÞ=ta. Applying the ¯ ¯ 4 Since SðρSÞ¼SðρRÞ, this completes the proof. constraint jI ðtaÞ − I ðtbÞj ≪ 1=jtb − taj , we find that max max For the unraveled channel N u (or unitary-dephasing the recovery operation can be decomposed into a time-local t channels, more generally), there is no need to apply the unitary circuit for sequences ða; bÞ satisfying 1 l where pm⃗ is the probability of a given quantum trajectory which we abbreviate as i Z2X3Y4 , with a group ρ ρ † and m⃗ ¼ Km⃗ Km⃗ is the density matrix for a single operation defined by the usual matrix multiplication. The trajectory. To prove this bound, we first purify the dynamics group is non-Abelian when accounting for the phase factors to a unitary operation on a combined reference, system, and il and has a total number of elements 4Nþ1. Given a environment, where the environment acts as a register to quantum state jψi on a d-dimensional Hilbert space, there record the quantum trajectories, is an associated subgroup of the unitary group UðdÞ called 041020-22 DYNAMICAL PURIFICATION PHASE TRANSITION INDUCED … PHYS. REV. X 10, 041020 (2020) a stabilizer group StabðjψiÞ defined as the set of unitaries be updated in a way that preserves the commutation U ∈ UðdÞ such that Ujψi¼jψi. Here, we are interested in relations of the Pauli group elements. More explicitly, the much more restricted set of stabilizer states, which are representing two elements of the Pauli group as equal to the set of states that are uniquely defined by their stabilizer group when restricted to the Pauli group Pi ¼ (ðzi; xiÞjri);i∈ f1; 2g; ðC2Þ SðjψiÞ ¼ StabðjψiÞ ∩ PN. More specifically, stabilizer ψ ⊂ P states j i are defined by Abelian subgroups S N such ðzi; xiÞ¼ðzi1;xi1; …;ziN;xiNÞ; ðC3Þ that −1 ∉ S, S has a minimal generating set of N linearly independent Pauli group elements, and, for every g ∈ S, we obtain gjψi¼jψi. Each such stabilizer group S has 2N elements. It is a convenient fact that stabilizer states are in a one-to- P1 þ P2 ¼ (ðz1 ⊕ z2; x1 ⊕ x2Þjr1 · r2); ðC4Þ one correspondence with the set of states that can be ⊗N X generated by acting on j0i with Clifford group circuits 2 2 2 4 C r1 · r2 ¼ r1 þ r2 þ gðz1j;x1j;z2j;x2jÞ mod ; [52]. The Clifford group for N qubits N is a subgroup of j N Uð2 Þ that has the defining property that each U ∈ CN maps elements of the Pauli group to other elements of the ðC5Þ Pauli group, i.e., UP U† ¼ P . A direct consequence of N N where we denote the Pauli group operation by , and this fact is that Clifford group circuits map stabilizer states þ g a; b; c; d is a function that takes 4 bits as input to stabilizer states. The full Clifford group is equal to the set ð Þ and outputs the power to which i is raised in the pro- of unitaries generated by Hadamard H, phase P, and CNOT μ μ gða;b;c;dÞμ μ I μ gates duct ab cd ¼ i a⊕c;b⊕d, where 00 ¼ ; 01 ¼ X; μ10 ¼ Z, and μ11 ¼ Y, i 11 e−iπ=4 0 H ¼ pffiffiffi ;P¼ ; gð0; 0;c;dÞ¼0;gð1; 1;c;dÞ¼c − d; 2 1 −1 0 eiπ=4 1 0 1 − 2 0 1 2 − 1 ⊕ gð ; ;c;dÞ¼dð cÞ;gð ; ;c;dÞ¼cð d Þ: UCNOTjs1;s2i¼js1;s1 s2i; This function encodes the single-site commutation relations ⊕ where is modulo-2 addition. Another important class of of the Pauli group. Given a generating set for a subgroup of operations that preserve stabilizer states includes measure- the Pauli group with M elements, as well as an additional ments of Hermitian Pauli operators. The set of quantum set of 2N − M linearly independent generators for the rest circuits generated by Clifford group circuits and measure- of the Pauli group, we can store all this information as a ments of Hermitian Pauli operators are known as stabilizer 2N × ð2N þ 1Þ-dimensional matrix over GFð2Þ, called a circuits. tableau representation for this subgroup. In Ref. [53], The Gottesman-Knill theorem states that, when stabilizer explicit algorithms are presented that take advantage of circuits act on stabilizer states, an efficient classical this tableau representation to evolve both pure and mixed algorithm exists to simulate their quantum dynamics stabilizer states in polynomial time in the number of qubits, [52]. The overall efficiency of this algorithm for a depth gates, and measurements applied to the system. t circuit with n measurements scales as O(ðt þ nÞN2) when implemented following the approach detailed by Aaronson and Gottesman [53]. In addition, partial traces, which map APPENDIX D: MIXED STABILIZER STATES pure stabilizer states to mixed stabilizer states, and entan- In this Appendix, we introduce some basic properties of glement of stabilizer states can also be efficiently computed mixed stabilizer states. If we define an Abelian subgroup in a time OðN3Þ [54,55,99]. The mathematical structure S ⊂ PN, such that −1 ∉ S, that is generated by M 041020-23 MICHAEL J. GULLANS and DAVID A. HUSE PHYS. REV. X 10, 041020 (2020) stabilizers Z¯ and associated flip operators X¯ to a complete for ρ consisting of an N × 2N matrix over GFð2Þ, whose i i ¯ ¯ generating set for PN by finding an additional set of rows are the binary vectors associated with fZ1; …; ZNg. 2 − ¯ ¯ … ¯ ¯ ðN MÞ operators fZMþ1; XMþ1; ; ZN; XNg, such that By restricting this matrix to the columns corresponding to c the whole generating set satisfies the usual Pauli algebra. If A , we can determine rA by performing row reduction on we think of the stabilizer group S associated with the mixed this N × 2jA¯ j matrix to put it into upper triangular form state as defining a stabilizer quantum-error-correcting code [54]. Such a procedure will generate a linearly independent 2 − c [81], then these additional ðN MÞ operators would be set of r¯A stabilizers that act nontrivially on A . The total referred to as logical operators in the code space. Here, the stabilizer group S is generated by this set of stabilizers, code space just refers to the subspace of the Hilbert space together with a generating set for SA, such that ρ on which acts trivially. N ¼ rA þ r¯A; thus, we arrive at the formula The combined unitary-projective measurement dynamics applied to the completely mixed state drives the system to a Sðρ Þ¼jAj − r ¼ r¯ − jAcj; ðE3Þ ¯ A A A mixed stabilizer state ρt with a set of generators Zi. If we fix a time t and initialize the system at t ¼ 0 in an arbitrary ¯ 3 ¯ where rA and rA can be computed in time OðN Þ using state in any of the code spaces with generators f Zig, then Gaussian elimination. these states will each be mapped under the dynamics to a Although the above algorithm can be applied to ρ unique state in the code space of t. As a result, all initial compute the bipartite entanglement of a given partition states in the code space can be recovered using a state- A ⊂ f1; …;Ng, there is a natural extension of this algo- independent unitary operation (fixed by the measurement rithm that can be used to efficiently compute the bipartite record but dependent on the gates and measurement entanglement of any contiguous region for a fixed ordering locations) that is a product of single-site unitaries that ¯ of sites, where contiguous is defined for periodic boundary collectively flip the generators Zi back to their sign in the conditions with respect to the chosen ordering [28,55]. initial state. First, we introduce the notation of the left lðgÞ and right rðgÞ end points of a stabilizer g, which are defined as the APPENDIX E: ENTANGLEMENT, ENTROPY, minimal and maximal sites on which g acts nontrivially. AND MUTUAL INFORMATION OF The algorithm proceeds by taking a tableau representation STABILIZER STATES for the generators of a stabilizer state with respect to the chosen ordering and performing row reduction on the entire In this Appendix, we describe methods to compute N × 2N matrix. This procedure effectively operates on the entropies of stabilizer states. The density matrix for a pure left end points of the stabilizers. In the second step, row stabilizer state has the explicit representation reduction is performed on the right end points, which aims to put the matrix into lower triangular form but with an 1 YN 1 X ρ I Z¯ g; E1 added constraint that one always eliminates right end ¼ 2N ð þ iÞ¼2N ð Þ “ ” i g∈S points by combining the stabilizer with a shorter stabi- lizer, where the length of a stabilizer g is defined as ¯ ¯ dðgÞ¼rðgÞ − lðgÞ. This second round of row reduction where fZ1; …; ZNg is a generating set for S. For pure states, the entanglement of a region A is simply the von Neumann preserves the left end points of the stabilizer generators and entropy of the reduced density matrix on A. From the results in a tableau matrix in the clipped gauge, which is expression for ρ, we can see that defined by the condition that, for every site x, the stabilizer generators satisfy X YrA 1 1 ¯ ρ ¼ Tr c ρ ¼ g ¼ ðI þ Z Þ; ðE2Þ ¯ ∶l ¯ ¯ ∶ ¯ 2 A A 2jAj 2jAj i nðxÞ¼jfZi ðZiÞ¼xgj þ jfZi rðZiÞ¼xgj ¼ ; ðE4Þ g∈S i A P with the sum rule n x 2N. The constraint in where S is a subgroup of S with the defining property that x ð Þ¼ A Eq. (E4) is an immediate consequence of the above row g is equal to the identity when restricted to Ac, i.e., jAcj ¯ ¯ reduction procedure. Such a representation does not I c ⊗ Tr c g ¼ 2 g, and fZ1; …; Z g is a generating A A rA uniquely fix the stabilizer generators, but it allows for an set for SA. From this expression, we can see that stabilizer efficient calculation of r¯A for any contiguous region in states have a completely degenerate entanglement spec- terms of the positions of the left and right end points trum, such that the von Neumann entropy is given by the [28,55], simple expression SðρAÞ¼−Tr½ρA log ρA ¼jAj − rA, where rA is the number of elements in a minimal generating 1 ρ ¯ ∶l ¯ ∈ & ¯ ∈ c set for SA. Thus, to compute the entanglement, it is Sð AÞ¼2 jfZi ðZiÞ A rðZiÞ A sufficient to find the order of S . This task can be A l ¯ ∈ c & ¯ ∈ accomplished by writing a reduced tableau representation or ðZiÞ A rðZiÞ Agj: ðE5Þ 041020-24 DYNAMICAL PURIFICATION PHASE TRANSITION INDUCED … PHYS. REV. X 10, 041020 (2020) As a result, by performing two steps of row reduction to put Given an ½N; k; d stabilizer code S with k>0 and the stabilizers into the clipped gauge, the entanglement can a set of geometric partitions of the qubits A ¼fAi∶Ai ⊂ be efficiently computed for all NðN − 1Þþ1 contiguous f1; …;Ngg, we define a code distance with respect to A, subregions by simply checking end-point positions of the stabilizers, which reduces the overhead of the entanglement dA ¼ minfN ∶∃g ∈ P ;gj c ¼ I;½g;S ¼0;g∉ Sg; ðF1Þ Ai N Ai calculation by a factor of OðN2Þ for each subregion. Ai Entanglement of mixed states is generically difficult to where PN is the Pauli group on N qubits and NA is the compute as it requires one to distinguish classical and number of elements in A. If no such Ai ∈ A exists, then we quantum correlations. However, entropies and mutual define dA ¼ maxAi NAi . The code distance d ¼ dP, where information of mixed stabilizer states can be efficiently P is the set of all partitions of f1; …;Ng. In general, computed using similar methods as described for pure computing the distance of an arbitrary stabilizer code is states. The entropy formula for a given subregion A has to expected to be exponentially hard in N. ¯ be updated because rA þ rA ¼ M (M is the number of For the stabilizer codes considered in Sec. VI, which are independent generators for the mixed stabilizer state), generated by an underlying 1D random circuit, it is natural to define the contiguous code length containing site x as ρ − − ¯ Sð AÞ¼jAj rA ¼jAj M þ rA; ðE6Þ l ≡ A x dAx , where x is the set of all contiguous regions of f1; …;Lg such that x ∈ A for every A ∈ A . We distin- e.g., S ρ N − M. Similar to pure states, we can define a i i x ð Þ¼ guish periodic and open boundary conditions by whether 1 clipped gauge for mixed states by performing left and then and L are considered neighbors. The average contiguous M right row reduction on the first rows of the tableau code length is defined as representation for ρ. For a given site x, instead of Eq. (E4), we have the identity 1 X l ¼ lx: ðF2Þ ¯ ¯ ¯ ¯ L x nðxÞ¼jfZi∶lðZiÞ¼xgj þ jfZi∶rðZiÞ¼xgj ≤ 2; P A closely related quantity called the linear code distance with the sum rule nðxÞ¼2M. x l ¼ min l was introduced by Bravyi and Terhal in Another difference between pure and mixed states in this min x x Ref. [101]. For a given partition A, we can determine representation is that it is not sufficient to know just the end whether there exists a logical Pauli group operator that lives points of the stabilizers in the clipped gauge to compute the on A by performing Gaussian elimination on the tableau entropy of contiguous subregions. For subregions A that do representation of a generating set for the logical operators not wrap around site N, there is a formula for the entropy: restricted to Ac, which takes a time at most polynomial in L [54]. Since the set of all contiguous regions of a 1D Sðρ Þ¼jAj − jfZ¯ ∶lðZ¯ Þ ∈ A & rðZ¯ Þ ∈ Agj: ðE7Þ A i i i geometry is polynomial in L, we can also compute the average code length for S in a time polynomial in L. The However, for contiguous regions that wrap around N, there average contiguous code length is an upper bound on the is the possibility that the left and right end points are both in code distance l ≥ l ≥ d. A, but the stabilizer has support outside of A. In this case, min In Fig. 9, we show the optimal code-averaged hli for the additional linear independence tests have to be performed 1 1 c 3 þ -dimensional model studied in Sec. VI. We perform on A , which can take time OðN Þ. As a result, for some the encoding step by starting from a completely mixed state contiguous regions, there is no advantage to working in the and running each circuit for a time t ¼ 4L. As a con- clipped gauge for the purposes of computing the subsystem vention, we define the code distance of an ½N;0 code (i.e., entropy of mixed states. a stabilizer state) as zero. With this convention, hli equals the probability that the system is in a mixed state (and thus APPENDIX F: CONTIGUOUS CODE LENGTH defines a code) times the average contiguous code length of those trajectories, which is why hli decays to zero in the A mixed stabilizer state is a normalized projector onto a pure phase. Deep in the mixed phase ξ=L ≪ 1, hli scales quantum-error-correcting code space. A shorthand notation subextensively as hli ∼ La ða ¼ 1=3 − 1=2Þ over this to identify the power of a code is ½N; k; d , which specifies range of sizes, whereas in the critical region of the mixed that the code is defined on N physical qubits, encodes k phase ξ=L ∼ 1, hli seems to scale as hli ∼ L. We also find l logical qubits, and can correct any error that acts on up to that the average linear code distance h mini (not shown) has ðd − 1Þ=2 physical qubits, where d is the code distance. the same scaling as the hli throughout the mixed phase. The code distance is defined as the minimal weight The extensive scaling of hli with system size near pc in this (number of nonidentity sites) of all possible Pauli group 1 þ 1-dimensional model gives us additional evidence in elements that commute with the stabilizer group S but are support of the bound on pc ≤ 0.1893 for the entanglement not contained in S. transition for qubits in 1 þ 1 dimensions [49]. 041020-25 MICHAEL J. 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