Passive correction of quantum logical errors in a driven, dissipative system: A blueprint for an analog quantum code fabric

Eliot Kapit, John T. Chalker, and Steven H. Simon

Phys. Rev. A 91, 062324 – Published 24 June 2015

Abstract

A physical realization of self-correcting quantum code would be profoundly useful for constructing a quantum computer. In this theoretical work, we provide a partial solution to major challenges preventing self-correcting quantum code from being engineered in realistic devices. We consider a variant of Kitaev's toric code coupled to propagating bosons, which induce a ranged interaction between anyonic defects. By coupling the primary quantum system to an engineered dissipation source through resonant energy transfer, we demonstrate a “rate barrier” which leads to a potentially enormous increase in the system's quantum-state lifetime through purely passive quantum error correction, even when coupled to an infinite-temperature bath. While our mechanism is not scalable to infinitely large systems, the maximum effective size can be very large, and it is fully compatible with active error-correction schemes. Our model uses only on-site and nearest-neighbor interactions and could be implemented in superconducting qubits. We sketch one such implementation at the end of this work.

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Received 19 August 2014
Revised 2 December 2014

DOI:https://doi.org/10.1103/PhysRevA.91.062324

Authors & Affiliations

Eliot Kapit, John T. Chalker, and Steven H. Simon

Rudolf Peierls Center for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom

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Vol. 91, Iss. 6 — June 2015

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Figure 1
Topological qubit array studied in this work, as described in Eq. (5). Here the black circles are spins, with the circuitry in the gray shaded regions depicting the five-body plaquette couplings $W_{♦ j} [V_{A 0} + g (a_{j}^{†} + a_{j})]$ between the spins and the type $A$ resonators (black solid squares) and the circuitry depicted in the white regions depicting the star couplings $W_{+ j} [V_{B 0} + g (b_{j}^{†} + b_{j})]$ to the type $B$ resonators (white squares with black borders). The black single and double lines depict the nearest-neighbor hopping between resonators of the same type. The propagating bosons in the resonator arrays induce ranged interactions in the plaquette and star terms. The model can still be solved exactly and has the same degeneracies and excitation structure of Kitaev's toric code (though there is now an interaction potential between anyons). By coupling the spins to an auxiliary shadow lattice of qubits engineered to have rapid decay rates (not shown in the figure), the system can passively correct quantum logic errors and function as a topologically protected quantum memory. Note that in a realistic device the many-body couplings would have to be engineered through a gadget construction, which requires additional qubits not shown here.
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Figure 2
Primary lattice from Fig. 1 with the passive error-correction module (shadow lattice) added. Each qubit (black circles) in the primary lattice is coupled to a group of shadow-lattice qubits (the nine boxed red circles) which have rapid decay rates that shrink or eliminate excitations in the primary lattice created by unwanted interactions with the environment. A group of shadow-lattice qubits may be shared between multiple primary qubits, and the couplings shown here are merely an example implementation where each group of nine shadow-lattice qubits is shared by four primary-lattice qubits. The shadow-lattice qubits within each group are tuned so that each one has a distinct excitation energy, with the energies chosen to match transition energies in the primary lattice. This resonance condition leads to rapid error correction while simultaneously minimizing additional errors generated by the shadow lattice itself.
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Figure 3
Potential landscape for an anyon pair. The plaquette at $r = 0$ is not drawn and is assumed to contain a single anyon; the heights of all other plaquettes correspond to the total energy $2 V - U (r)$ of placing a second anyon on that plaquette. Note that as $U ≪ V$ the minimum energies on the plot are still large. A spin sits at each shared corner and by flipping the spin anyons can be created, annihilated, or exchanged between plaquettes. For each flip which brings anyons closer together, the system loses energy, and with a properly tuned shadow lattice, these flips will occur rapidly at a rate $Γ_{R} (δ E)$ . As spin flips which move anyons further apart occur at a much slower primary error rate $Γ_{P}$ , the probability for a local pair of anyons to separate widely after creation is suppressed by a factor $\propto \prod_{n = 1}^{n_{\max}} [Γ_{R} (δ E_{n}) / Γ_{P}]$ , where $n_{\max}$ is the average maximum number of steps it would take to separate anyons such that the potential differences for further steps are too small for the shadow lattice to act upon. For $Γ_{R} ≫ Γ_{P}$ , this can lead to an enormous increase in the lifetime of quantum information encoded in the system's topological ground-state degeneracy.
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Figure 4
Anyon removal as an error source. In (a), spins $i$ and $k$ have flipped, creating four anyons in the adjoining plaquettes. The shadow lattice will rapidly correct these errors, and if spin $i$ or $k$ flips as shown in (b), the remaining anyon pair can be quickly eliminated, returning the system to its ground state. If, however, spin $j$ flips first as in (c), the remaining anyon pair will require three further operations to eliminate and is thus much more likely to lead to a quantum logic error than in case (b). To reduce the likelihood of this process, we add a second set of shadow-lattice qubits which are tuned to be close to resonance with the energy removed in process (b) but not process (c), ensuring that process (b) will occur much more rapidly, making process (c) rare.
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Figure 5
Repair functions $Γ_{R} (δ E)$ used in our simulations. The three curves plot the repair rates generated by shadow lattices tuned to pull separated anyons back together for $- μ_{A} = 0.1 J$ (thick black), $0.2 J$ (dashed red), and $0.4 J$ (thin blue), with $J = 3 g$ and all energies expressed in units of $g$ . The primary repair terms which annihilate anyons are targeted at energies $δ E \sim - 60 g$ , so have negligible influence on processes in the region plotted. The precise parameters of each shadow lattice are detailed in Appendix pp1.
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Figure 6
Quantum-state lifetime $τ$ (in units of $g^{- 1}$ , where $g$ is the driven plaquette and/or star coupling to the resonator array) of a logical state encoded in the topological quantum degeneracy of an $L \times L$ toroidal implementation of our qubit array. From top to bottom, the systems simulated have propagating boson gaps $- μ / J = 0.4$ , 0.2, and 0.1; the shadow-lattice parameters used for passive error correction are detailed in Table 1. In each plot we display the average lifetime $τ$ computed for (bottom to top curves) primary error rate $Γ_{P} = {3, 1.5, 1, 0.75, 0.5} \times 10^{- 4} g$ . The best and worst average lifetimes come from systems which differ by mere factors of 3 in $L$ and 6 in $Γ_{P}$ , but over $10^{4}$ in $τ$ itself. Note also that the best case studied ( $18 \times 18, - μ / J = 0.1, Γ_{P} = 0.5 \times 10^{- 4} g$ ) has an average lifetime around 7000 times longer than any of the individual qubits. Due to the finite length scale $l = \sqrt{- J / μ}$ of the interaction potential, the error correction in each case saturates around $L_{c} \approx 7 l$ . Beyond $L = L_{c}$ increasing the size of the lattice simply adds more potential anyon sources without significantly increasing the probability of anyons being recaptured once they reach the random-walk regime (a separation of more than $\sim 2 l$ for the parameters considered). Each data point for $Γ_{P} = 0.5 \times 10^{- 4} g$ was generated from 400 error simulations.
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Figure 7
Comparison of different primary repair strengths for $μ / J = - 0.1$ and $Γ_{P} = 10^{- 4}$ . The curves are computed using the secondary repair parameters listed in Table 1 and varying primary repair strengths. We plot the results for the primary repair parameters in Table 1 (blue circles), a 50% increase the values of $Ω_{1}, Ω_{2}, Γ_{S 1}, Γ_{S 2}$ (gold squares), a 100% increase (green diamonds), and a 150% increase (red, triangles). The optimal values of the primary repair terms depend on the size of the lattice, but making them too strong eventually becomes counterproductive and reduces the lifetime, as it reduces the selectivity of the error-correction process discussed in Fig. 4. Each data point shown is the average of 400 error simulations.
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Figure 8
Implementation of edges in our passive error-correction scheme. As in the surface code [5], nonperiodic boundary conditions are implemented by truncating four-body operators to three-body terms; along the top or bottom edges, stars are truncated, and along the left or right edges plaquettes are truncated. The figure depicts the upper left corner of a larger rectangular patch of quantum computing fabric.
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Figure 9
Quantum-state lifetime (in units of $g^{- 1}$ ) of an $L \times L$ plaquette implementation of our system with physical edges, using the parameters described in the text and calculated using the defect-tracking algorithm in Appendix pp1. A linear potential gradient $δ V (r) = - ω_{9} | y - L / 2 |$ is applied to recapture single anyons created at either edge; $ω_{9}$ is the lowest shadow-lattice energy in Table 1 and $y$ is the vertical coordinate. The curves are computed from primary error rates $Γ_{P} = {3, 1.5, 1, 0.75, 0.5} \times 10^{- 4} g$ (bottom to top curves) as in Fig. 6, again using $J = 3 g, μ / J = - 0.1$ , and the shadow-lattice parameters in Table 1. The even-odd effects seen at lower error rates stem from $y = L / 2$ lying on or between plaquette centers. To mitigate further edge effects, the ranged interaction $U (r)$ was calculated assuming that the $L \times L$ primary lattice was embedded at the center of a much larger ( $100 \times 100$ ) resonator array. The parameters chosen here almost certainly do not represent the optimal choice for $δ V (r)$ or the shadow-lattice energies. To reduce computation time, the values of $τ$ with $Γ_{P} = 0.75 \times 10^{- 4} g$ and $0.5 \times 10^{- 4}$ were computed from 400 error simulations per data point (compared to 900 for the other values) and thus have a slightly higher uncertainty.
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Figure 10
Three-, four-, and five-body gadgets, implemented in transmon qubits. The Josephson junctions in blue boxes are the qubits of the primary lattice, and the circuits in red boxes are high-energy gadget degrees of freedom which are integrated out to produce three- and four-body couplings in perturbation theory. Appropriately tuned, the five-body gadget will generate both four- and five-body terms ( $- σ_{1}^{x} σ_{2}^{x} σ_{3}^{x} σ_{4}^{x} [J_{4}^{'} + J_{5} (a + a^{†})]$ ), implementing the qubit Hamiltonian Eq. (5). All devices share a common (bridged) ground and are coupled by small Josephson junctions with time-dependent flux biases, as described in Eq. (B1) in Appendix pp2.
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Figure 11
Eigensystem of the three-body gadget derived from the construction in Fig. 10. The blue circles plot the eigenvalues $σ_{1}^{x} σ_{2}^{x} σ_{3}^{x}$ of the eigenstates and the purple squares plot the energy in units of $κ$ . The degeneracy can be made exact by careful tuning; values used in this simulation were (all in units of $κ$ ) $κ_{3} = 1, ω_{R} - ν = 4, κ_{c} ≃ 0.64$ , and $c_{3} ≃ 0.3$ . There are eight low-lying states, with the excited states all separated from them by a large excitation energy and thus inaccessible in the low-energy limit.
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Figure 12
Eigensystem of the four-body gadget constructed from chaining together three-body gadgets. The low-energy Hamiltonian is $H = - J_{4} σ_{1}^{x} σ_{2}^{x} σ_{3}^{x} σ_{4}^{x}$ , where $J_{4} ≃ 2 J_{3}^{2} / (ω_{Q 5} - ν^{'})$ . The blue circles plot the eigenvalues $σ_{1}^{x} σ_{2}^{x} σ_{3}^{x} σ_{4}^{x}$ of the eigenstates and the purple squares plot the energy in units of $κ$ . Higher-order interactions can be also generated through this mechanism. Here the counterterm $κ_{c}$ is identical to the three-body case, $c_{5} = 2 c_{3}$ , and $ω_{Q} - ν^{'} = 2 κ$ .
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