PHYSICAL REVIEW X 4, 041039 (2014)

Measurement-Free Topological Protection Using Dissipative Feedback

Keisuke Fujii,1,2,3 Makoto Negoro,1 Nobuyuki Imoto,1 and Masahiro Kitagawa1 1Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan 2The Hakubi Center for Advanced Research, Kyoto University, Yoshida-Ushinomiya-cho, Sakyo-ku, Kyoto 606-8302, Japan 3Graduate School of Informatics, Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto 606-8501, Japan (Received 12 June 2014; revised manuscript received 3 October 2014; published 1 December 2014) Protecting quantum information from decoherence due to environmental noise is vital for fault-tolerant quantum computation. To this end, standard quantum error correction employs parallel projective measurements of individual particles, which makes the system extremely complicated. Here, we propose measurement-free topological protection in two dimensions without any selective addressing of individual particles. We make use of engineered dissipative dynamics and feedback operations to reduce the entropy generated by decoherence in such a way that quantum information is topologically protected. We calculate an error threshold, below which quantum information is protected, without assuming selective addressing, projective measurements, or instantaneous classical processing. All physical operations are local and translationally invariant, and no parallel projective measurement is required, which implies high scalability. Furthermore, since the engineered dissipative dynamics we utilize has been well studied in quantum simulation, the proposed scheme can be a promising route progressing from quantum simulation to fault-tolerant quantum information processing.

DOI: 10.1103/PhysRevX.4.041039 Subject Areas: Condensed Matter Physics, Quantum Physics, Quantum Information

I. INTRODUCTION in conventional QEC are very challenging (see Fig. 1). A recent study [15] has revealed that parallel projective A standard approach to protect quantum information measurements of individual 108 qubits are required every from decoherence is the use of the celebrated quantum hundred nanoseconds to maintain quantum coherence for error correction (QEC) [1–3]. It conventionally employs factorization of 1024-bit composite numbers. (The total projective measurements, classical information processing, amount of information measured is 1 petabit= sec.) While and feedback operations with selective addressing of monolithic architectures, quantum dots [15], and super- individual particles. Based on the standard paradigm of conducting qubits [16,17] on a chip exhibit promising QEC, fault-tolerant quantum computer architectures have scalability, macroscopic measurement devices coupled with been designed with the additional ability to operate individual qubits for parallel projective measurements universal quantum gates [4–14]. might introduce other sources of decoherence, thereby However, there exist severe problems that have to be overcome for the realization of standard QEC. The pro- limiting this approach. On the other hand, distributed jective measurements and classical processing utilized in architectures, consisting of modules comprising a small standard QEC have to be much faster than the coherence number of qubits, connected with optical channels, allow both accurate manipulations and measurements inside the time of quantum systems, which is extremely challenging – in experiments. If classical processing depends on the size local modules [18 25]. However, the entangling operation of the system, it ultimately limits both the speed and the between separate local modules using flying photons takes a size of quantum computers. From a theoretical viewpoint, it long time due to photon loss. Hopefully, these problems will is highly nontrivial to establish an error threshold theory be overcome within the conventional paradigm by a break- including the classical system to control quantum com- through in the development of accurate manipulations and puters. In practice, fast and reliable parallel projective parallel projective measurement technology. In the mean- measurements of the massive numbers of qubits employed time, we should not stop searching for a novel way toward robust and scalable protection of quantum information. Here, we propose a new paradigm, measurement-free topological protection (MFTP) of quantum information Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- using dissipative dynamics, paving a novel way toward bution of this work must maintain attribution to the author(s) and fault-tolerant quantum computation. We unify the quantum the published article’s title, journal citation, and DOI. system that is to be protected and a controlling classical

2160-3308=14=4(4)=041039(11) 041039-1 Published by the American Physical Society FUJII et al. PHYS. REV. X 4, 041039 (2014)

FIG. 1. MFTP versus standard topological QEC (STQEC). MFTP with discrete-dissipative feedback operations, which are implemented by local and translationally invariant physical operations (left). A STQEC with projective measurements and classical processing (right). Since MFTP does not require any selective addressing, its scalability is evident. system in a framework without assuming parallel projec- [32]. The surface code, in which a qubit is located on each tive measurements, selective addressing, or instantaneous edge of an L × L square lattice as shown in Fig. 2(a),is classical processing. Specifically, we devise a way to stabilizedQ by the face andQ vertex stabilizer operators, correct errors on topological quantum codes with restricting A Z and B X , respectively. Here, Z f ¼ i∈Ef i v ¼ j∈Ev j i our controllability to translationally invariant nearest- and Xj denote Pauli operators on the ith and jth qubits, and neighbor unitary operations and single-spin dissipative Ef and Ev indicate the sets of four edges surrounding a face dynamics in a two-dimensional system as follows. The f and adjacent to a vertex v, respectively. The error classical system is coupled with the quantum system, in syndromes fafg and fbvg are defined as sets of eigenvalues which quantum information is stored, in order to extract the of the face and vertex stabilizers, respectively, which are syndrome of errors. With this information, the classical used to identify X and Z errors. These errors are assumed to system is cooled down by an engineered dissipative occur on each qubit with independent and identical error dynamics. (See Ref. [3] for a comprehensive review of probability p, for simplicity. quantum error correction and an engineered dissipation In standard topological quantum error correction aimed at quantum memory.) The obtained low-energy state (STQEC) [3,31], the error syndrome is extracted to the in the classical system is utilized with a feedback operation classical world through parallel projective measurements. to correct errors on the quantum information in the According to the error syndrome, we use a classical quantum system. The lifetime of the stored quantum algorithm, minimum-weight perfect matching (MWPM) information is shown to be improved polynomially in [33], which tells us the location of the errors. By virtue of the size (linear length) of the system, provided that the the locality and translational invariance of the surface code, error probability is smaller than a threshold value. the error threshold of the STQEC is very high (∼1%) even Since the proposed topological protection does not when using only the nearest-neighbor two-qubit gates to require any selective addressing or parallel projective extract the error syndrome [10–12]. However, the necessity measurements, MFTP enables us to easily achieve scal- of parallel projective measurements and the classical ability in various physical systems (see Fig. 1). Further- processing may limit the effectiveness of STQEC. more, engineered dissipative dynamics utilized for If measurement-free QEC is allowed, both of these topological protection has been well studied in the context limiting factors can be eliminated in fault-tolerant quantum of quantum simulation [26–29]. MFTP serves as a computation. In Refs. [34,35], measurement-free QEC with promising route to progress from quantum simulation to a nontopological code was investigated, where errors are fault-tolerant quantum information processing. corrected by unitary dynamics with selective addressability of the boundary qubits. It is well known that the projective II. TOPOLOGICAL PROTECTION measurements can be replaced with preparations of fresh We consider a 2D many-body quantum system that ancillas (by dissipation) followed by controlled-unitary encodes quantum information, for simplicity, using the operations. However, if we straightforwardly apply this surface code [30,31] [the lower layer in Fig. 2(a)]. The strategy to the surface code, the classical processing, i.e., proposed scheme can also be applied straightforwardly to MWPM, is too complex to be implemented by unitary other local stabilizer codes such as topological color codes dynamics. The unitary dynamics for MWPM is far from

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FIG. 2. The system and dissipative dynamics for MFTP. (a) The quantum (lower layer) and classical (upper layer) many-body systems for quantum information storage and its feedback controls, respectively. The Z errors occur on the qubits colored by red. The associated vertex stabilizers with the eigenvalue −1 are denoted by black squares. The four classical spins adjacent to each vertex interact according to the eigenvalues of the vertex stabilizers. The wrong-sign plaquettes with bv ¼ −1 are shown by gray squares on the upper layer. The double layer is not necessarily required, but different degrees of freedom of the particles located on one layer can be employed. (b) The effect of plaquette interaction. A correct-sign plaquette with bv ¼þ1 favors an even number of −1’s, where a chain of −1’sis energetically hard to terminate. The wrong-sign plaquette favors an odd number of −1’s, where a chain of −1’s terminates easily. (c) The competing interactions reveal the location of errors: two wrong-sing plaquettes are connected with a short length. local and translationally invariant, which completely dimin- In this sense, the proposed model achieving topological ishes the merit of using the surface code. This is also the protection of quantum information in nonequilibrium can case for topological quantum error correction by a sophis- be regarded as a quantum analog of dynamic memory, ticated local update rule, which requires a complex classical which periodically refreshes the capacitor charge to store ancilla system [36]. It is nontrivial to design the system to information reliably. be topologically protected with local and translationally We utilize a classical 2D many-body system, in addition invariant physical operations, which are feasible enough to the quantum system, as an ancilla to allow feedback to implemented in an actual experiment. The proposed operations of discrete type [the upper layer in Fig. 2(a)]. For scheme, MFTP, makes active use of dissipative dynamics to simplicity, we explain the feedback operation for the Z reduce the entropy of the quantum system in such a way error correction only. (The X error correction can also be that quantum information is topologically protected with done in a similar way.) According to the error syndrome local and translationally invariant operations, thus enabling fbvg, we cool (dissipate the energy) the classical system us to fully utilize the advantage of the surface code. down to a sufficiently low but a finite temperature T under a Hamiltonian III. DISCRETE-DISSIPATIVE FEEDBACK X Y X FOR TOPOLOGICAL PROTECTION HðfbvgÞ ¼ −J bv ui − h ui; ð1Þ ∈ Recently, extensive research has been conducted on v i Ev i engineering dissipative dynamics to simulate open quan- 1 tum systems [26] or to prepare quantum states of interest where ui ¼ denotes the classical spin located on edge [37–40]. Unfortunately, the local and translationally invari- i and J and h indicate the coupling constant and the ant dissipations toward thermal equilibrium at finite tem- magnetic field strength, respectively. Here the “classical” perature cannot protect quantum information due to no-go spins refers to the qubits without their phase coherence. We theorems of self-correcting quantum memory [41,42]. call this Hamiltonian the 2D random-plaquette gauge While extensive effort is being made, such as the toric- model with magnetic fields. Then the equilibrium configu- boson model [43], a self-correcting quantum memory has ration is obtained and put back into the quantum system by not been achieved by dissipative dynamics under a fixed transversal controlled-Z (CZ) operations between the local Hamiltonian system. A quantum version of a non- classical (controls) and quantum (targets) systems. volatile magnetic storage device in equilibrium, such as a The above discrete-dissipative feedback process suc- hard disk drive, seems to be hard to achieve at finite ceeds in correcting errors in the surface code through the temperature. Not only dissipative dynamics but also the following progression. The location of errors in the surface time dependence of the Hamiltonian, which drives the code can be identified by finding a minimum path system into nonequilibrium with a feedback mechanism, connecting pairs of incorrect eigenvalues bv ¼ −1 [31]. would be key ingredients to achieve topological protection. The first term in Eq. (1), which we call the plaquette

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FIG. 3. Cooling dynamics and topological error correction. The location of the Z errors in the quantum system and wrong-sign plaquettes are shown by red and black squares, respectively (top left), where the error probability is p ¼ 0.05. Equilibrium configurations are shown for βJ ¼ 0.1 (top middle), 0.5 (top right), 0.8 (bottom left), and 2.3 (bottom middle), where −1 spins are shown by gray squares. An equilibrium configuration at low temperature is put back into the quantum system. The remaining Z errors after the feedback are shown by blue squares (bottom right). Most of the errors are corrected, but some errors, shown by blue squares, remain. These are corrected by subsequent cycles. interactions, imposes the boundary conditions of a chain of In order to confirm the above observation, we perform −1’s as shown in Fig. 2(b). A correct-sign plaquette with numerical simulations of the classical cooling process bv ¼ 1 energetically favors an even number of −1’s, where under the Hamiltonian HðfbvgÞ using the Metropolis a chain of −1’s is hard to terminate. Awrong-sign plaquette method. Figure 3 illustrates equilibrium configurations at with bv ¼ −1 energetically favors an odd number of −1’s, βh ¼ 0.1, 0.5, 0.8, and 2.3, where β is the inverse temper- where a chain of −1’s terminates easily. On the other ature and J ¼ h is adopted, for example. The remaining Z hand, the second term in Eq. (1), which we call magnetic errors after the feedback operation are also shown. Most of fields, favors configurations of fewer −1s’. If the plaquette the errors are corrected within one cycle of MFTP, as shown interactions are chosen to be strong enough to connect pairs in Fig. 3 (lower right-hand panel). However, some errors of wrong-sign plaquettes in the presence of the magnetic still remain, which are attributed to the excitations in the fields, these two competing interactions are expected to plaquette interactions that result from a finite temperature reveal the locations of the Z errors at a sufficiently low effect. Such errors are corrected in the following MFTP temperature [see Fig. 2(c)]. In Appendix A, we show that, cycles and/or suppressed by the logarithmic scaling of the by choosing the strength of plaquette interactions such that plaquette interactions 4J=ð2hÞ¼α log L. 4J=ð2hÞ¼α log L with a constant α, the logical error We further investigate the logical error probability with probability per step decreases polynomially in the system varying the system size L and the physical error probability p. size L, as long as the physical error probability p is below Specifically, we choose α ¼ 1 and utilize the thermal equi- a threshold value. Since the correlation length of the system librium state of a temperature T ¼ 1=β ¼ −2h=ln½p=ð1−pÞ scales like Oðlog LÞ, the relaxation time is expected to on the Nishimori line [44], where the thermal fluctuation and depend on the system size L polylogarithmically (see physical error probability are balanced. In the limit of large J Appendix A). (i.e., large α), the discrete-dissipative feedback under the

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γ FIG. 4. Numerical results. The effective decay rate eff is estimated by numerical simulations for the physical error probabilities 0 01 0 02 … 0 05 8 10 12 16 20 γ p ¼ . ; . ; ; . and the system size L ¼ ; ; ; ; . (a) The effective decay rate eff as a function of the physical error probability p. From top to bottom L ¼ 8; 10; 12; 16; 20. (b) The effective decay rate γeff as a function of the system size L. From bottom to top p ¼ 0.01; 0.02; 0.03; 0.04; 0.05.

Nishimori temperature achieves an optimal decoding of the We utilize simultaneous two-qubit gates on neighboring γ surface code. The effective decay rate eff of the stored particles of different species, such as controlled-NOT information under topological protection is calculated by (CNOT) gates between A and B qubits, for example. In −γ t fitting the logical error probability to 3ð1 − e eff Þ=4 with t addition, we use a dissipative operation, specifically T1 being the time step. Since two logical qubits are encoded relaxation or incoherent pumping, on the classical spins. into the surface code, the fitting is chosen such that the The procedure in each MFTP cycle for the Z error logical error probability converges to 3=4 in the large t correction is as follows (see also Fig. 6). (i) Initialize B limit. (The fidelity of a maximally mixed state of two and B’ to j0i. (ii) Perform CNOT gate operations between A 1 4 γ qubits is = .) The effective decay rate eff is plotted as (controls) and B (targets) to extract the error syndrome. functions of the physical error probability p and the size L Perform CNOT gate operations between B (controls) and B’ in Figs. 4(a) and 4(b), respectively. We further analyze the (targets) to copy the syndrome. (iii) Simulate the cooling scaling of the effective decay rate with respect to the linear dynamics in a digitalized way by using the stabilizer length L of the system and the physical error probability pumping [26–29] [see Fig. 5(b)]. (iv) Perform CZ gate Γ ½ElogðpÞþF 0 82–1 15 ’ p. We obtain eff ¼DL with D ¼ . . , operations between A (controls) and A (targets) to apply E ¼ 1.54 0.05,andF ¼ 4.06 0.18. This indicates that the lifetime of the quantum information is improved polynomially in the size of the system L, which is in good agreement with the theoretical analysis made in Appendix A. We observe that the threshold for physical error probability with α ¼ 1 is at least as high as 5.0%. The extrapolation of Fig. 4(a)or equivalently the condition E logðpÞþF<0 provides the threshold value around 6.0%–7.0%.

IV. IMPLEMENTATION AND FEASIBILITY OF MFTP We now consider a physical implementation of MFTP. We consider a system with two layers, each of which consists of three species of particles, as shown in Fig. 5(a) (left-hand side). The A qubits and A’ spins are the quantum and classical systems considered, respectively. The B and C qubits are the ancillas for the Z and X error syndrome ’ ’ FIG. 5. Physical implementation of MFTP. (a) The double-layer extractions, respectively. The B and C spins are used to system for MFTP. Each 2D layer consists of three species of mediate the syndrome-dependent plaquette interactions. particles (left). The different degrees of freedom of the particles Instead of the double-layer system, particles on a single on a single 2D layer can also be utilized (right). (b) The stabilizer layer, which have different degrees of freedom, can be pumping scheme for a digital simulation of the cooling process utilized as shown in Fig. 5(a) (right-hand side). (see Appendix B).

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the environment, we obtain Markovian digitalized cooling dynamics,

† τLP τLF m TrE½VðtÞρA ⊗ ρEVðtÞ ≃ ½e e ρA;

where ρA and ρE indicate density matrices of the ancilla system and the environment, respectively. LP and LF denote the Lindblad superoperators [46] with respect to the Markovian decays under HP and HF, respectively. These are realized by using single-qubit dissipative dynamics and two-qubit gates, as shown in Fig. 5 [26–29] (see Appendix B). We now discuss the requirements of the physical parameters in the experiments. The decoherence rate of the quantum system A, B, and C is denoted by Γ. The FIG. 6. One MFTP cycle for the Z error correction. (i) The ’ strength of the dissipative operation on the classical spins B qubits and B spins are initialized to extract and copy the error A’,B’, and C’ is denoted by γ ∼ ∥H ∥. We also define a syndrome, respectively. (ii) The error syndrome is extracted from int coupling strength κ between the qubits and spins, which the A qubits to the B qubits. The extracted error syndrome is 1 κ copied to the B’ spins. (iii) By using the copied error syndrome, limits the gate time to = . Furthermore, we utilize a two-qubit gates, and single-qubit dissipative dynamics, the Markov approximation, J; h ≫ γ, and the Trotter-Suzuki γτ2 ≪ 1 γτ2 ≪ 1 cooling process under the Hamiltonian HðfbvgÞ is simulated expansion imposes J and h . By setting in a digitalized way [see Fig. 5(b)]. (iv) The obtained equilibrium J; h ∼ 10γ, Jγτ2 ∼ 10−1, and hγτ2 ∼ 10−1, the time interval configuration, parts of which reveal the location of the Z errors, is τ of each digitalized cooling step becomes τ ∼ 10−1=γ. put back into the A qubits to correct the Z errors. According to the numerical simulations at least up to the system size L ¼ 400, the cooling process is accomplished 2 4 the feedback operation. A similar procedure for the X error within ∼10 –10 Monte Carlo steps, each of which correction is also performed with a basis change using physically takes a relaxation time of ∼1=γ for local spin Hadamard operations. These MFTP cycles for the X and Z flipping. Thus, the cooling time tcool required for one ∼ 102–104 γ error corrections are repeated. MFTP cycle is estimated to be tcool = . The cooling process in step (iii) is simulated in a Since the cooling time tcool is divided into digitalized digitalized way, which we call digitalized cooling, using cooling steps of time interval τ ∼ 10−1=γ, the number of τ ∼ 103–105 stabilizer pumping [3,26–29]. We can realize the syndrome- repetitions is m ¼ tcool= . In each digitalized dependent plaquette interactions automatically by using the cooling step, only a few unitary operations are required five-body interactions described by since most of the unitary gates are commutable and operate X Y X simultaneously. Accordingly, in addition to the cooling ~ − B0 A0 − A0 time t , the unitary gate operations take ∼m=g.Asa H ¼ J Zv Zi h Zi ; ð2Þ cool v i∈Ev i result, the time taken by one MFTP cycle is κ ∼ 102 γ 103 κ since the error syndrome is copied onto the B’ spins in tcycle ¼ tcool þ m= = þ = : step (ii). Note that the classical spins A’,B’, and C’ are In order for MFTP to work, the physical error probability denoted as if they are qubits, for simplicity of the notation. −Γ p 1 − e tcycle H ¼ for the quantum system has to be as small By introducing an interaction int between the ancilla 10−2 classical spins and the environment, the time evolution of as . (When both X and Z errors are corrected using the − ~ same classical layer, the physical error probability effec- ≡ iðHþHintÞt the total system is given by VðtÞ e . If the tively doubled.) Hence, we obtain requirements on the coupling strength with the environment is sufficiently small, −4 −6 −5 physical parameters, Γ=γ ≲ 10 –10 and Γ=κ ≲ 10 . and the correlation time is sufficiently short, then the time The former and latter conditions indicate that the speeds evolution of the system can be regarded as a Markovian of dissipation and unitary operations have to be 104–106 decay. Then, the time evolution is divided into short 105 digitalized cooling steps of interval τ ¼ t=m: and times faster than the decoherence time of the quantum system, respectively. If these conditions are

−i H H =2 τ −i H H =2 τ m satisfied, the effective decay rate per cycle is reduced to V t ≃ e ð Pþ int Þ e ð Fþ int Þ ; −8 ð Þ ½ ∼10 (the physical decoherence rate is reduced 6 orders of P Q P magnitude) assuming the scaling for L ¼ 400. where H ≡−J ZB0 ZA0 , H ≡−h ZA0 , and the P v v i∈Ev i F i i Here, we discuss the feasibility of the above conditions Trotter-Suzuki expansion [45] is employed. By tracing out in the case of a solid-state system with nuclear and electron

041039-6 MEASUREMENT-FREE TOPOLOGICAL PROTECTION USING … PHYS. REV. X 4, 041039 (2014) spins for quantum and classical systems, respectively. establishes a promising means of progressing from quan- State-of-the-art chemistry and nanotechnology research tum simulation to fault-tolerant quantum computing. has demonstrated the possibility of engineering materials with nuclear and electron spins arranged on a lattice. In ACKNOWLEDGMENTS solids such as diamond, the T2 relaxation time of nuclear spins exceeds 1 sec even if it is located near an electron The authors thank S. Nishida, Y. Morita, H. Tasaki, and spin [47]. The electron spins can be initialized within B. Yoshida for valuable discussions. This work was 1=γ ∼ 100 nsec if they have appropriate optical transitions. supported by the Funding Program for World-Leading The hyperfine interaction between electron and nuclear Innovative Research and Development on Science and spins and the dipolar interaction between electron spins Technology (FIRST), MEXT Grant-in-Aid for Scientific for two-qubit gates are both typically on the order of Research on Innovative Areas 20104003 and 21102008, κ ∼ 1–100 MHz. These parameters fulfill the above con- the MEXT Global COE Program, and JSPS Grant-in-Aid 1 for Research Activity Start-up 25887034. ditions. In such a case, tcycle ¼ msec, and the lifetime of quantum information under MFTP with L ¼ 400 becomes 105 sec, i.e., more than 1 day. APPENDIX A: ANALYSIS OF THE Other degrees of freedom in natural or artificial atoms, LOGICAL ERROR PROBABILITY for example, such as the motional and internal degrees of The energy penalty for the boundary mismatching is 4J, freedom in trapped atoms [26–29] or superconducting flux since they are always created as a pair of excitations with qubits coupled with microwave cavities [16,17], can also be respect to plaquette interactions. On the other hand, a utilized. connected chain of −1’s of a length r costs an energy 2hr due to the magnetic fields. Thus, the pair of two wrong-sign V. CONCLUSION AND DISCUSSION plaquettes is connected in a typical case when the distance We propose MFTP, which is a novel way to protect between two wrong-sign plaquettes is shorter than ≡ 4 2 quantum information from decoherence using dissipative rcor J=ð hÞ, which gives a typical length of the system operations without parallel projective measurements or (more precisely, the maximum correlation length at zero selective addressing. We show that local and translationally temperature). If errors occur contiguously, and the distance invariant physical operations without any selective address- between two wrong-sign plaquettes becomes larger than ing and measurements achieve topological protection of rcor, then the boundary mismatching is energetically quantum information against errors on the code state. A favored due to the energy penalty of magnetic fields. fully fault-tolerant analysis should be made, further taking This means that the spins of the classical system search into account those imperfections in the syndrome extraction pairs of wrong-sign plaquettes inside an area whose vertical and the engineered dissipative dynamics. To this end, the and horizontal lengths are typically given by rcor. classical ancilla system, instead of using a fully quantum In the following, we bound the logical error probability system, of a finite temperature without necessity of phase after the feedback operation through a two-step argument. coherence would have an advantage with respect to its First, we consider a necessary condition: the energy land- robustness. Moreover, for fault-tolerant quantum compu- scape of the classical system HðfbvgÞ is given appropri- tation, we have to engineer the boundaries of planer surface ately such that the spin configuration at zero temperature codes in order to accommodate many logical qubits and to can correct the error appropriately. This is not sufficient for perform logical gates between them. our purpose, since we utilize a spin configuration at finite Recently, quantum memory (surface codes) coupled with temperature. Thus, second, we bound the logical error another quantum or classical system, such as classical probability for the given appropriate energy landscape fields, spins, or a bosonic reservoir, has been investigated taking into account the finite temperature effect. extensively to increase the lifetime of quantum information For a given error chain Ce, we define a minimum-path [43,48,49]. Among them, the automaton decoder [48], chain Cm, which connects pairs of wrong-sign plaquettes where anyonic excitations (boundaries of error chains) with a shortest path (i.e., the solution of MWPM). When are annihilated by an attractive force given by classical errors are dense, the error and minimum-path chains Ce þ potential fields made by the excitations themselves, shares Cm would constitute a connected chain of a length longer the idea with our proposal. The automaton decoder would than rcor. In such a case, the error correction fails even if we also be made measurement-free if there was a feasible way have the spin configuration at zero temperature, resulting in to implement the classical fields and their update rules. an inappropriate energy landscape. To calculate such a The dissipative dynamics has recently been well studied failure probability perror, we consider an area whose vertical in quantum simulation. Actually, the key ingredients of and horizontal lengths are given by rcor. The probability MFTP have already been demonstrated in quantum sim- that the error and minimum-path chains Ce þ Cm constitute ≥ ulation [27–29]. The proposed model, an active use of the a connected chain of length l rcor can be calculated by engineered dissipative dynamics for topological protection, following the method provided in Ref. [31]:

041039-7 FUJII et al. PHYS. REV. X 4, 041039 (2014) X Xl l X Xl=2 l k 1 − l−k k 1 − l−k −2βhðl−2kÞ perror < CðlÞ p ð pÞ pex ¼ CðlÞ p ð pÞ e pgs l≥r k>l=2 k ≥ 0 k Xcor l rcor k¼ 2 < CðlÞ2lpl= ; X Xl=2 l −2βh l−k −2βh k l≥r < CðlÞ ½e ð1 − pÞ ðp=e Þ ≥ 0 k lXrcor k¼ where CðlÞ is the number of self-avoiding walks of length l < CðlÞ2lpl=2; l−1 ≥ inside this area, which is bounded as CðlÞ < 3 × 4 . l rcor Since we can begin such a connected chain of length l ≥ r at any one of L2 lattice sites, the total failure probability where p ≥ 1 is the population of the ground state, and the cor 2 gs 1 − 1 − L ≃ 2 −2β is at most ð perrorÞ L perror. [In STQEC with temperature is chosen to be e h ¼ p=ð1 − pÞ.Thecon- −2β MWPM, rerror is replaced with the system size L. This dition e h ¼ p=ð1 − pÞ, the so-called Nishimori line [44], reproduces the calculation made in Ref. [31], leading to the indicates that the thermal fluctuation with respect to the threshold value independent of the system size. The fact magnetic field and the physical error probability are bal- that rerror ¼ OðLÞ implies that communication over the anced. As a result, we can obtain the same scaling of the whole system is employed in the classical processing for failure probabilities perror and pex to the leading order in the MWPM, which requires a time OðLÞ in the presence of presence of the thermal fluctuation below the Nishimori line. parallelism. By using an elaborated renormalization algo- Similarly to the previous case, the connected chain of length rithm, we can reduce the time complexity for this commu- ≥ 2 l rcor can begin at any one of L lattice sites, and hence, nication to Oðlog LÞ [50], where nonlocal operations, 2 the total failure probability in this case is at most L pex. which are prohibited in the present setup, are still required. Finally, we consider the latter case (ii). Since the number See also Table I.] 2 of connected errors in this area is at most rcor= , the Since we utilize a spin configuration at finite temper- 4 − 2 2 boundary mismatching costs at least J hðrcor= Þ ature, we further take into account excitations that originate energy penalty. Thus, the probability of the excitation from the finite temperature effect to calculate the logical causing the boundary mismatching can be given by error rate. There are two types of excitations: (i) excitations without boundary mismatches and (ii) excitations with −4βJþ2βhðrcor=2Þ −βhrcor boundary mismatches. In the former case (i), while the pbm ¼ e pgs

TABLE I. A comparison of the performances of the related topological protection schemes based on the surface codes. The types of protections, active and passive, indicate the noise models and how to suppress errors. The active type mainly assumes probabilistic errors on the surface code, which are corrected by a feedback process. The passive type assumes thermal excitations on the surface code (the Toric code model), which are suppressed by an energy penalty coupling with an ancilla system or a higher dimensionality.

Coupling Memory Decoding Type of operations Model Type strength time time employed (in 3D) Dynamics Our proposal with J ¼ Oðh log LÞ Active Oðlog LÞ polyðLÞ poly logðLÞ Local and translationally Nonequilibrium invariant driven by feedback Our proposal with J ¼ OðhLÞ Active OðLÞ expðLÞ polyðLÞ Local and translationally Nonequilibrium invariant driven by feedback Surface code in 4D without Passive Constant expðLÞ No Nonlocal translationally Thermal equilibrium error correction [31] invariant Surface code in 2D without Passive Constant Constant No Local and translationally Thermal equilibrium error correction [51] invariant Toric-boson model [43] Passive polyðLÞ polyðLÞ No Local and translationally Thermal equilibrium invariant Surface code in 2D with Active Not defined expðLÞ polyðLÞ Nonlocal and no Nonequilibrium MWPM [31] translationally invariant driven by feedback Surface code in 2D with a Active Not defined expðLÞ polylogðLÞ Nonlocal but having Nonequilibrium renormalization decoder [50] self-similarity driven by feedback

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Summing all of these failure probabilities, the logical error probability is given by

2 2 2 plogical ¼ L perror þ L pex þ L pbm X 2 8 2 2 6l l= 1 − rcor=

Suppose the typical length rcor is chosen to be rcor ¼ α log L with a constant α. The failure probability converges 1 − ≤ −4=α 36 to zero in the limit of a large L if p=ð pÞ e = . FIG. 7. The energy landscape of the ancilla system. At a high α 1 −4β −3 For example, with ¼ and 2, this condition reads the temperature, such as e J ¼ 10 , the system decays to the ≃ 5 1 10−4 3 7 10−3 threshold value pth . × and . × , respec- thermal equilibrium state rapidly. Then the temperature is further tively. We note that the above calculation is far from tight, reduced to the Nishimori temperature. The relaxation time scales since CðlÞ is overestimated substantially. Furthermore, in polylogrithmically in the size L of the system at least up to many cases the boundary mismatches do not cause a fatal L ¼ 400, where at most 104 Monte Calro steps are enough for failure, since the remaining errors would be corrected in the equilibration with unsophisticated schedulings of annealing. following steps. Thus, the true threshold value is expected to be much higher, as shown in the numerical simulation. temperature, as shown in Fig. 7. Then, such a remaining The relaxation time is an important factor for the error may be corrected in the next round. proposed scheme, since the physical error probability p Based on these arguments, we expect that the delay time τ τ ∼ is increased exponentially during the delay of the feedback. for the feedback scales like poly logðLÞ.Thethreshold The relaxation time is typically given as a polynomial for the decoherence rate of the quantum system scales like Γ ∼ 1 function of the correlation length of the system. [In a th =poly logðLÞ. In practice, it should be better to τ accomplish an error correction procedure as fast as possible critical system, the typical time scale of the system crit and τ z rather than waiting a slow convergence to the thermal equi- the correlation length rcrit are related by crit ¼ rcrit, with z being the dynamical critical exponent. The present classical librium state. The optimization of the annealing schedule of the classical ancilla system is an intriguing future issue. system HðfbvgÞ is not critical, and the relaxation time is z Finally, we consider a large J limit, where J is scaled as expected to be a lower order polynomial function than rcor.] In the present case, the typical length of the system depends OðhLÞ. In such a case, rcor scales like OðLÞ. Thus, the α discrete-dissipative feedback at the Nishimori temperature on the system size only logarithmically, rcor ¼ log L. Thus, the relaxation time required to obtain a thermal provides an optimal decoding, which is slightly better than equilibrium state or a metastable state enough to execute the decoding with MWPM [52,53]. An exponential sup- error correction is expected to be a polylogarithmic func- pression of the logical error can also be achieved under this tion of the system size L. scaling. However, if the coupling strength scales like ∼ We numerically confirm that the classical system rapidly J OðhLÞ, the relaxation time (Monte Carlo steps) would decays to the thermal equilibrium state when the inverse also scale polynomially in the size of the system. It is temperature is chosen to be e−4βJ ∼ 10−3. Specifically, still open whether or not the fast scheduling of classical the equilibration is accomplished within several hundred dissipative dynamics, possibly using metastable states, can Monte Carlo steps at least up to the system size L ¼ 400. be managed to suppress the logical error exponentially. We This is because metastable states relatively easily climb up summarize the performance of the proposed scheme the energy barrier and find the pairs at such a temperature, compared with other related models, including decoding as shown in Fig. 7. Then the temperature of the classical methods based on the explicit parallel measurements and system is further reduced to the Nishimori temperature, the classical processing in Table I. where the unpaired spin chains are also shrunk due to the energy penalty e−βh. In numerical simulations, with less APPENDIX B: DIGITAL SIMULATION than 104 Monte Carlo steps, we can prepare the classical OF THE COOLING PROCESS ancilla system of the system size at least L ¼ 400. At least The Lindblad superoperators L and L for Markovian up to L ¼ 400, the relaxation time scales like a polylogar- P F decays under the Hamiltonian H and H are given by ithmic function of the system size L. With a small P F X probability, the classical ancilla system is still a metastable L ρ −γ ηˆ†ηˆ ρ ρηˆ†ηˆ − 2ηˆ ρηˆ† state due to the energy barrier 4βJ at the low temperature. P ¼ − ð v v þ v v v vÞ ∈ 0 However, error correction using such a metastable state vXB − γ ηˆ ηˆ†ρ ρηˆ ηˆ† − 2ηˆ†ρηˆ does not spoil error correction, since the length of unpaired þ ð v v þ v v v vÞ; ðB1Þ spin chains is suppressed due to the energy penalty at low v∈B0

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X Y L ρ −γ0 σˆ þσˆ −ρ ρσˆ þσˆ − − 2σˆ −ρσˆ þ F ¼ − ð i i þ i i i i Þ ˆ − ξv ¼ UXv I − Zv Zi U ¼ XvðI − ZvÞ=2 ¼ σv ; i∈A0 X i∈Ev − γ0 σˆ −σˆ þρ ρσˆ −σˆ þ − 2σˆ þρσˆ − þ ð i i þ i i i i Þ: ðB2Þ i∈A0 which is the single-qubit lowering operator for the spins Q A0 B0 A0 and hence implemented using T1 relaxation or incoherent The operator ηˆ ¼ X ∈ ðI − Z ∈ Z Þ=2, where i ∈ ~ v i Ev v i Ev i pumping. Thus, by combining the operation F and U, the E means that one of the edges in E is chosen randomly, v v Q dissipative operation LP can be realized as lowers the energy with respect to −JZB0 ZA0 . The v i∈Ev i − operators σˆ þ ¼j1ih0j and σˆ ¼j0ih1j are the raising and τL ~ τL~ i i i i e P ρ ¼ FUe P ρ; lowering operators for the ith qubit, respectively. 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