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Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes

Christopher Chamberland, Kyungjoo Noh, Patricio Arrangoiz-Arriola, Earl T. Campbell, Connor T. Hann, Joseph Iverson, Harald Putterman, Thomas C. Bohdanowicz, Steven T. Flammia, Andrew Keller, Gil Refael, John Preskill, Liang Jiang, Amir H. Safavi-Naeini, Oskar Painter, and Fernando G.S.L. Brandão

PRX Quantum 3, 010329 – Published 23 February 2022

Abstract

We present a comprehensive architectural analysis for a proposed fault-tolerant quantum computer based on cat codes concatenated with outer quantum error-correcting codes. For the physical hardware, we propose a system of acoustic resonators coupled to superconducting circuits with a two-dimensional layout. Using estimated physical parameters for the hardware, we perform a detailed error analysis of measurements and gates, including cnot and Toffoli gates. Having built a realistic noise model, we numerically simulate quantum error correction when the outer code is either a repetition code or a thin rectangular surface code. Our next step toward universal fault-tolerant quantum computation is a protocol for fault-tolerant Toffoli magic state preparation that significantly improves upon the fidelity of physical Toffoli gates at very low qubit cost. To achieve even lower overheads, we devise a new magic state distillation protocol for Toffoli states. Combining these results together, we obtain realistic full-resource estimates of the physical error rates and overheads needed to run useful fault-tolerant quantum algorithms. We find that with around 1000 superconducting circuit components, one could construct a fault-tolerant quantum computer that can run circuits, which are currently intractable for classical computers. Hardware with 18 000 superconducting circuit components, in turn, could simulate the Hubbard model in a regime beyond the reach of classical computing.

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Received 21 December 2020
Revised 3 November 2021
Accepted 26 January 2022

DOI:https://doi.org/10.1103/PRXQuantum.3.010329

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum algorithms Quantum benchmarking Quantum error correction Quantum gates Surface code quantum computing Topological quantum computing

Quantum Information, Science & Technology

Authors & Affiliations

Christopher Chamberland ^1,2,*, Kyungjoo Noh¹, Patricio Arrangoiz-Arriola^1,†, Earl T. Campbell^1,†, Connor T. Hann ^1,3,†, Joseph Iverson^1,†, Harald Putterman^1,†, Thomas C. Bohdanowicz^1,2, Steven T. Flammia¹, Andrew Keller¹, Gil Refael^1,2, John Preskill^1,2, Liang Jiang^1,4, Amir H. Safavi-Naeini^1,5, Oskar Painter^1,2, and Fernando G.S.L. Brandão^1,2

¹AWS Center for Quantum Computing, Pasadena, California 91125, USA
²IQIM, California Institute of Technology, Pasadena, California 91125, USA
³Department of Physics, Yale University, New Haven, Connecticut 06511, USA
⁴Pritzker School of Molecular Engineering, The University of Chicago, Illinois 60637, USA
⁵Department of Applied Physics and Ginzton Laboratory, Stanford University, Stanford, California 94305, USA

^*[email protected]
^†These authors contributed equally.

Popular Summary

Building a universal fault-tolerant quantum computer is one of the greatest challenges of the 21st century. In doing so, good qubits along with reliable gates performing all the entangling operations between the qubits is a necessity. Furthermore, fault-tolerant quantum error correction will be required to ensure that all logical qubits and gates acting on the qubits fail with the very low probabilities required by large-scale quantum algorithms. In this paper, we propose a full stack analysis for building a universal fault-tolerant quantum computer and provide a resource cost analysis for implementing quantum algorithms using our architecture. In particular, we propose a quantum-computing architecture built on dissipative cat qubits with the property that one type of error (bit-flip errors) are exponentially suppressed. We then show how such qubits can be realized using acoustic phononic modes coupled to nonlinear elements called asymmetrically threaded superconducting quantum interference devices. We then go on to propose a fault-tolerant architecture with such qubits that does not require long-range gate connectivity and which exploits the noise bias to achieve low hardware requirements for its implementation. In doing so, we propose a new fault-tolerant bottom-up protocol for preparing Toffoli magic states at the physical hardware level. The bottom-up protocol is then combined with a novel top-down magic state distillation scheme to achieve the desired low failure rates required by algorithms. Such Toffoli magic states, along with thin-stripped surface codes for encoding our logical qubits can be used alongside lattice surgery methods to implement a universal gate set. Combining all of the above, we show that with 32 000 superconducting circuit components, the Hubbard model can be simulated in a regime, which is beyond reach by classical computers.

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Vol. 3, Iss. 1 — February - April 2022

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Figure 1
Bloch sphere of the cat qubit. The codewords $| 0 ⟩$ , $| 1 ⟩$ and the $| \pm ⟩$ states are indicated on the $Z$ and $X$ axes, respectively, along with their Wigner function representations (shown for $α = 2$ ).
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Figure 2
Hardware implementation of the repetition- and surface-cat codes. In the 2D grid on the right, yellow circles represent data qubits where the logical information is encoded, and gray circles represent ancilla qubits, which are used to measure the stabilizers and extract error syndromes. Both data and ancilla qubits are encoded as Schrödinger cat states of localized acoustic modes of PCDRs, and are stabilized through a driven-dissipative two-phonon interaction with an engineered reservoir. This stabilization strongly biases the noise, suppressing $X$ errors and increasing $Z$ errors. The letter $R$ in each plaquette represents the reservoir, which is implemented with a capacitively shunted ATS (the “buffer” resonator), a bandpass filter, and an open waveguide. This circuit is shown inside the “RESVR” box in the left panel and has a single nongrounded terminal, marked with a white circle on the edge of the box. All resonators surrounding each reservoir in the layout diagram in the center panel connect to this one physical terminal. The green circles represent an additional acoustic mode used to measure the cat qubits in the $X$ basis with the aid of a transmon, which is represented by a white square. Altogether, five PCDRs are connected to each reservoir: four as active qubits, and one for readout.
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Figure 3
Dimensionless loss $κ_{1} / κ_{2}$ , as given by Eq. (10), as a function of buffer decay rate $κ_{b}$ and storage intrinsic energy relaxation time $T_{1, i} = 1 / κ_{1, i}$ , assuming fixed $ω_{a} / 2 π = 2.16 GHz$ and $| α |^{2} = 8$ . We label the corresponding quality factor $Q_{i} = ω_{a} / κ_{1, i}$ on the upper horizontal axis, and mark the $κ_{b}$ value used in our proposal with the white dashed line. The points corresponding to REGIME 1, REGIME 2, and REGIME 3 are indicated with a circle, a square, and a star, respectively. Future innovations in the multiplexed stabilization scheme may allow for larger bandwidths, which would relax these requirements proportionally.
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Figure 4
Multiplexed stabilization and crosstalk mitigation. (a) Frequency multiplexing. Because the desired couplings $(g_{2} {\hat{a}}_{n}^{2} {\hat{b}}^{†} e^{i Δ_{i} t} + H.c.)$ are detuned by different amounts, photons lost to the environment via the buffer have different frequencies. When the corresponding emitted buffer-mode photons (green lines) are spectrally well resolved, $| Δ_{n} - Δ_{m} | ≫ 4 | α |^{2} κ_{2}$ , the modes are stabilized independently. Dissipation associated with photon emissions at frequencies inside the filter passband (the yellow box, with bandwidth $4 J / 2 π = 100$ MHz) is strong, while dissipation associated with emission at frequencies outside the passband is suppressed. (b),(c) Crosstalk suppression. Red lines in (b) denote photon-emission frequencies associated with various correlated errors, calculated for the specific storage-mode frequencies plotted in (c). The mode frequencies are deliberately chosen so that all emissions associated with correlated errors occur at frequencies outside the filter passband (no red lines fall in the yellow box). In other words, Eqs. (18) and (19) are simultaneously satisfied for any choices of the indices that lead to nontrivial errors in the cat qubits. See Appendix pp2 for further details.
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Figure 5
(a) Circuit illustration of a $d = 5$ repetition code embedded in our ATS layout. As explained in Fig. 2, the yellow vertices correspond to the data qubits and the gray vertices to the ancillas, whereas green circles are readout modes and the white squares are transmon qubits, which we use for $X$ -basis measurements (see Appendix pp7 and Fig. 40 for more details). The pink semicircles are used to illustrate the $X_{i} X_{i + 1}$ stabilizer of the repetition code. We also label each cnot gate by the corresponding time step in which it is applied. (b) Circuit illustration of a $d_{x} = 3$ by $d_{z} = 5$ thin rotated surface code. The pink and blue plaquettes correspond to the $X$ - and $Z$ -type stabilizers, respectively, with the numbers indicating the time steps in which the cnot gates are applied. Measurements of $X$ -type stabilizers detect $Z$ errors whereas measurements of $Z$ -type stabilizers detect $X$ errors. (c) Key illustrating the different components of the repetition- and surface-code lattices. cnot gates are used to couple qubits connected to the same ATS.
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Figure 6
Logical $Z$ failure rates for the repetition code, for a variety of values of the code distance $d$ . We use the circuit-level noise model described in Sec. 3 with $κ_{ϕ} = 0$ and $n_{th} = 0$ . The $X$ -basis measurement error rates are obtained from Table 3 with three parity measurements. The number of syndrome measurement rounds $r$ for each distance is obtained using the STOP algorithm described in Appendix pp9. The dashed green line is used as a stand-in for comparison with the logical memory error rates and corresponds to the function $0.3025 \sqrt{κ_{1} / κ_{2}}$ , which is a quarter of the total $Z$ failure rate of a physical cnot gate (see Table 2).
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Figure 7
Total logical failure rate per code cycle for various repetition-code distances and values of $| α |^{2}$ with fixed $κ_{1} / κ_{2} = 10^{- 5}$ , $κ_{ϕ} = 0$ , and $n_{th} = 0$ . Therefore, the $| α |^{2} = 8$ data points correspond to REGIME 3. The logical $X$ and $Y$ failure rates were computed analytically (to leading order) using the noise model presented in Sec. 3 while taking into account all malignant and benign fault locations. For $| α |^{2} = 8$ , the lowest achievable total logical error rate is $2.7 \times 10^{- 8}$ per code cycle using $d = 9$ .
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Figure 8
Logical $Z$ failure rates for the rotated surface code with $d_{x} = 3$ and varying $d_{z}$ . We use the circuit-level noise model described in Sec. 3 with $κ_{ϕ} = 0$ and $n_{th} = 0$ . The $X$ -basis measurement error rates are obtained from Table 3 with five parity measurements. We point out that $κ_{1} / κ_{2} = 10^{- 5}$ , $κ_{1} / κ_{2} = 10^{- 4}$ , and $κ_{1} / κ_{2} = 10^{- 3}$ correspond to cnot failure rates of $3.8 \times 10^{- 3}$ , $1.2 \times 10^{- 2}$ , and $3.8 \times 10^{- 2}$ . The simulations were done by performing $d_{z}$ rounds of noisy syndrome measurements followed by one round of perfect syndrome measurement.
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Figure 9
Crosstalk errors due to multiplexed stabilization. Phononic modes that are connected via a shared ATS experience correlated Pauli $Z$ errors due to micro-oscillation (see Appendix 19). Every pair of two data qubits that are shared by the same ATS (hence aligned vertically) undergoes a correlated $Z$ error with a probability $p_{double}$ . Also, every triple of two data qubits and an ancilla qubit that measures an $X$ -type surface-code stabilizer undergoes a correlated $Z$ error with a probability $p_{triple}$ , where the $Z$ error on the ancilla qubit manifests as a flipped outcome of the corresponding $X$ -type stabilizer.
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Figure 10
Logical $Z$ failure rates for a $d_{x} = 3$ and $d_{z} = 11$ thin surface code in the presence of the residual crosstalk errors [given in Eq. (42)] arising from the coherent micro-oscillations and the circuit-level noise model of Sec. 3. The $X$ -basis measurement error rates are obtained from Table 3 with five parity measurements. We compute the logical $Z$ -error rates for different values of $g_{2}$ shown in the legend, and compare such results to the case where the crosstalk errors are not present.
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Figure 11
Space-time diagram of memory and lattice-surgery processes using a thin, rotated surface code with boundaries. Pink (the left and right sides) represents boundaries where $Z$ strings can terminate. Blue (the fore and rear sides) represents boundaries where $X$ strings can terminate. We show examples of $Z$ strings: when traveling in a spatial direction they represent physical $Z$ errors and when traveling in the vertical time direction, they represent errors on $X$ stabilizer measurements. We show only $Z$ strings that are closed loops or terminate on suitable boundaries. These can be regarded as the final $Z$ strings (after matching) including physical and measurement errors combined with recovery operations. In the case of memory, a logical $Z$ error occurs whenever a $Z$ string propagates between two topologically disconnected red boundaries. When performing lattice surgery to measure the $X_{L 1} \otimes X_{L 2}$ logical operator between two patches, an additional failure mechanism is possible. If a $Z$ string propagates between two red boundaries disconnected in the time direction then we have a timelike $Z$ error. Computationally, this flips the outcome of the $X_{L 1} \otimes X_{L 2}$ measurement. Such processes are exponentially suppressed by increasing the measurement distance $d_{m}$ .
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Figure 12
Top: Toffoli gate injection that implements a Toffoli gate by consuming a $| TOF ⟩$ resource state. The Clifford corrections depend on the three measurement outcomes and are given in Eqs. (44) to (46). All qubits and gates are implemented at the logical level. Bottom: these circuit identities illustrate how to convert a teleportation circuit into a Pauli-based computation. We present five equivalent circuits showing how to convert from a cnot followed by measurements with $m$ cnot gates into a Pauli-based computation that can be realized by lattice surgery. Circuit 1 to 2: we insert the identity. Circuit 2 to 3: we have replaced the highlighted box with a multiqubit $Z^{\otimes m + 1}$ measurement. Circuit 3 to 4: we add a single qubit $X$ measurement before we discard the qubits. Circuit 4 to 5: we use the $X$ measurement to replace the cnot gates with classically controlled $Z$ gates. A similar identity holds with the cnot direction reversed and the roles of $X$ and $Z$ interchanged. Applying the identities of the bottom figure in the $m = 1$ case to the top figure yields a Pauli-based Toffoli gate injection procedure. We make use of the $m > 1$ case in Sec. 7 and Appendix 41b.
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Figure 13
Circuit for our entire BUTOF protocol. The first step (shown in the dashed blue box) consists of preparing the state $| ψ_{1} ⟩ = \frac{1}{\sqrt{2}} (| 100 ⟩ + | 111 ⟩)$ . The preparation of the states $| 0 ⟩_{L}$ and $| 1 ⟩_{L}$ are described in Appendix 38. The next step consists of measuring $g_{A} = X_{A} {CNOT}_{B, C}$ . If the measurement outcome on the ancilla is $- 1$ , a $Z_{A}$ correction is applied to the output state. Note that at this stage, error correction is not applied to the data block. The first two steps are enclosed within the dashed red box. We label the output state of the first two steps as $| ψ_{out} ⟩$ . Lastly, the measurement of $g_{A}$ is repeated $(d - 1) / 2$ times for a distance $d$ repetition code. The ED blocks correspond to one round of stabilizer measurements of the repetition code. If any of the measurement outcomes of error detection (ED) or ancillas are nontrivial, the protocol is aborted and begins anew.
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Figure 14
(a) Implementation of the $g_{A}$ measurement (for a distance $d = 5$ repetition code) compatible with our ATS layout and lattice-surgery implementation for universal quantum computation described in Sec. 7. All operations are performed respecting the connectivity constraints of the ATSs and use the fewest possible ancilla qubits for preparing the GHZ state necessary for the fault-tolerant measurement of $g_{A}$ . (b) Equivalent circuit for the implementation of (a).
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Figure 15
(a) Total logical $Z$ failure rate for preparing a $| TOF ⟩$ state using the fault-tolerant BUTOF protocol described in this section. (b) Acceptance probabilities for preparing $| TOF ⟩$ states using the fault-tolerant protocols described in this section. (c) Decomposition of the logical $Z$ errors for a $d = 7 | TOF ⟩$ state prepared using the fault-tolerant protocol described in this section. As can be seen, from all seven possible combinations of logical $Z$ errors, a logical $Z$ error on block $A$ is more likely by several orders of magnitude. All numerical simulations were performed by setting $n_{th} = 0$ , $κ_{ϕ} = 0$ , and using the circuit-level noise model described in Sec. 3 with $| α^{2} | = 8$ .
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Figure 16
Our TDTOF protocol uses a trio of quantum error-correction codes each using $n = 8$ qubits to encode $k = 2$ logical qubits. The top circuit shows the $3 n = 16$ qubits and labels the $X$ stabilizers and logical operators for these codes. Each of the $3 k = 6$ qubits is actually encoded again into a thin surface code. The bottom circuit shows the $3 k$ logical qubits encoded by this code. The crucial and exotic property of these codes is that they have a transversal a CCZ gate: applying the eight CCZ gates as shown in the top circuit, results in the two logical CCZ gates shown in the bottom circuit. See Appendix 39 for proof details.
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Figure 17
Schematic diagram of an ATS. Two junctions with Josephson energies $E_{J, 1}$ , $E_{J, 2}$ are connected in parallel, forming a SQUID. The SQUID loop in turn is “split” in the middle by a linear inductor with inductance $L_{b}$ , effectively forming two loops on either side of the inductor. The magnetic fluxes $ϕ_{ext, 1}$ and $ϕ_{ext, 2}$ threading the left and right loops, respectively, are controlled via externally applied, time-dependent currents $I_{1} (t)$ , $I_{2} (t)$ that are buffered to ground in the vicinity of the loops using on-chip fluxlines.
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Figure 18
Calculation of $g_{2}$ . (a) Schematic summary of our method for calculating $g_{2}$ . A PCDR, connected in parallel to a buffer resonator that is formed by shunting an ATS with a capacitance $C_{b}$ , is synthesized as a simple Foster network with the same admittance function $Y_{m} (ω)$ as the original piezoelectric structure. The Foster network consists of a parallel combination of an inductance $L_{a}$ and a capacitance $C_{a}$ , in series with a “coupling capacitance” $C_{g}$ . In turn, the linear components of the buffer resonator $L_{b}$ and $C_{b}$ are lumped together with the mechanical Foster circuit, leaving the nonlinear part of the ATS potential as an additional circuit element that we label by $U (ϕ)$ in the diagram. The linear network is then diagonalized and the vacuum fluctuation amplitudes $φ_{a}$ and $φ_{b}$ of the storagelike and bufferlike eigenmodes are used to calculate $g_{2}$ . (b) Dependence of $g_{2}$ on the buffer resonator frequency $ω_{b}$ and impedance $Z_{b}$ . The $g_{2}$ curves peak at the storage-mode frequency $ω_{a}$ where the modes are maximally hybridized. Inset: $g_{2}$ plotted as a function of $Z_{b}$ for a fixed $ω_{b}$ , showing the $5 / 2$ power-law dependence.
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Figure 19
Filter design. (a) Schematic of the filtering setup. A bandpass filter centered at frequency $ω_{b}$ is placed in between the output of the buffer resonator and an open waveguide with characteristic impedance $Z_{0}$ and phase velocity $v_{ϕ}$ . Photons that are transmitted through the filter enter the open waveguide and are irreversibly lost. (b) Circuit diagram showing the normal modes $a$ and $b$ and their connection to the filter described by an admittance function $Y (ω)$ . (c) Detailed circuit diagram for the filter structure, which consists of a main filter chain with $N$ “unit cells” followed by a taper section with $N_{t}$ cells, terminated at the end with a load resistance $Z_{0}$ that accurately models the open waveguide at the output port. Every cell of the filter has frequency $ω_{f}$ and impedance $Z_{f}$ , and neighboring cells are coupled capacitively with capacitances $C_{c}^{(i, i + 1)}$ . The coupling capacitance $C_{κ}$ between the buffer resonator and the first filter cell is defined separately for generality. (d) Top: coupling capacitances plotted as a function of cell index $i$ for tapered [ $(N, N_{t}) = (10, 3)$ ] and uniform [ $(N, N_{t}) = (13, 0)$ ] filters. The tapered structure, found automatically by a Nelder-Mead optimizer, is characterized by a rapid increase in $C_{c}$ near the end of the structure. Bottom: typical filter response, here shown as the real part of $Y (ω)$ for tapered and uniform filters. The response of the uniform structure shows multiple sharp peaks, each corresponding to a standing-wave resonance of the structure, whereas the tapered response is relatively flat throughout the filter passband. In effect, the taper allows propagating waves to be transmitted to the open waveguide over a broad bandwidth.
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Figure 20
Exploring the filter design space. (a) Dependence of $κ_{b, eff} (Δ)$ and $κ_{2} (Δ)$ on the coupling capacitance $C_{κ}$ between the $b$ mode and the first filter resonator. Here we fix $ϵ_{p, 0} = 0.015$ , $ω_{f} / 2 π = 2.55 GHz$ , $Z_{f} = 500 Ω$ , $ω_{b} = ω_{f} - 2 J$ , $Z_{b} = 1 k Ω$ , $C_{c} = 3.0 fF$ , and $(N, N_{t}) = (10, 3)$ . As $C_{κ}$ increases, we observe two regimes: an “undercoupled” regime $C_{κ} ≪ C_{c}$ characterized by a sharply peaked $κ_{2} (Δ)$ , where the narrow $b$ mode filters the dissipation process, and an “overcoupled” regime $C_{κ} ≫ C_{c}$ where $κ_{2}$ saturates and becomes asymmetric. In this latter regime, the $b$ mode strongly hybridizes with the first filter cell. For large enough $C_{κ}$ , their normal-mode frequencies shift outside of the filter passband, forming bound resonances that are visible as sharp peaks to the left of the passband in some of the curves. The optimal value is $C_{κ} = C_{c} = 3.0 fF$ , where $κ_{2} (Δ)$ is maximized and flat, is shown in red. Note that at this optimal coupling, $κ_{b, eff} = 4 J$ (gray dashed line). Note also that the adiabatic condition $g_{2} < η κ_{b} / 2 α$ is not respected for several of the plots shown, as $g_{2}$ is fixed. (b) Dependence of $κ_{b, eff} (Δ)$ and $κ_{2} (Δ)$ on the characteristic filter impedance $Z_{f}$ . In order to keep the filter bandwidth $4 J$ constant, increasing $Z_{f}$ requires decreasing $C_{c}$ , and to keep $g_{2} < η κ_{b} / 2 α$ (adiabatic threshold), increasing $Z_{f}$ requires increasing $ω_{f}$ (which decreases $g_{2}$ due to the larger detuning between the $a$ and $b$ modes). The values used for the plotted curves are $ω_{f} / 2 π = 2.4, 2.5, 2.55, 2.6, 2.7 GHz$ , $C_{c} = 10, 4.5, 2.7, 1.7, 1.2 fF$ , $ω_{b} = ω_{f} - 2 J$ , and $(N, N_{t}) = (10, 3), (10, 3), (10, 3), (14, 6), (14, 6)$ . Larger values of $Z_{f}$ required larger $N_{t}$ to compensate for the larger impedance mismatch to the $50 Ω$ line. We also fix $Z_{b} = 1 k Ω$ here. The optimal value is $Z_{f} = Z_{b} / 2 = 500 Ω$ , which produces a flat $κ_{2} (Δ)$ curve (shown in red). Also note that at this optimal value, $κ_{b, eff} = 4 J$ (gray dashed line). Inset: $g_{2}$ and $η κ_{b} / 2 α$ corresponding to each of the simulations for the different values of $Z_{f}$ , plotted as a function of $ω_{b}$ , showing the adiabatic constraint $g_{2} < η κ_{b} / 2 α$ is satisfied (here $α = \sqrt{8}$ and $η = 1 / 5$ ).
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Figure 21
Behavior of $κ_{1} / κ_{2}$ in the large $Z_{b}$ limit. Top left panel: $g_{2}$ , $κ_{b}$ , and $κ_{2}$ plotted as a function of $Z_{b}$ . The latter two rates are averages over the middle of the filter passband, $ω \in [ω_{b} - J, ω_{b} + J]$ . The detunings $δ = ω_{b} - ω_{a}$ corresponding to each simulation are also indicated on the horizontal axis. Each result is obtained by optimizing the filter for each value of $Z_{b}$ , following the procedure outlined in Appendix 41. We observe that all of these rates remain practically constant, in particular, $κ_{b} = 4 J$ . Top right panel: single-phonon relaxation rate $κ_{1}$ plotted as a function of $Z_{b}$ and $δ$ . This relaxation rate $κ_{1} = κ_{1, i} + κ_{1, p}$ includes two contributions: the intrinsic loss $κ_{1, i}$ of the resonator, which here we assume has a fixed intrinsic quality factor $Q_{a, i} = ω_{a} / κ_{1, i} = 10^{9}$ , and the Purcell loss $κ_{1, p}$ due to its coupling to the buffer resonator. This latter rate has a contribution due to radiation into the waveguide (which is vanishingly small due to the strong filter suppression), and an important contribution $\sim (g / δ)^{2} κ_{b, i}$ due to the intrinsic decay of the buffer resonator itself, which we assume has $Q_{b, i} = 10^{6}$ . This loss channel is not suppressed by the filter. However, it can be mitigated by increasing the detuning $δ$ . Indeed, at large values of $δ$ , $κ_{1}$ asymptotes to $κ_{1, i}$ (gray dashed line). We also show the loss parameter $κ_{1} / κ_{2}$ plotted in red, where $κ_{2}$ is averaged over the filter band, which also asymptotes to a lower bound as $Z_{b}$ and $δ$ become large. Bottom panels: loss spectra $κ_{1} / κ_{2} (Δ)$ shown for a few selected values of $Z_{b}$ . The gray shading indicates the regions where the adiabatic condition $g_{2} < η κ_{b} / 2 α$ is not satisfied. These regions roughly correspond to the frequencies outside of the passband. Here $α = \sqrt{8}$ and $η = 1 / 5$ .
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Figure 22
Multiplexed stabilization. (a) Comparison of stabilization for $Δ_{n} = 0$ and $| Δ_{n} - Δ_{m} | ≫ 4 | α |^{2} κ_{2}$ . Wigner plots are shown of two storage modes after evolution under the master equation $\dot{\hat{ρ}} = - i [\hat{H}, \hat{ρ}] + κ_{b} D [\hat{b}]$ , with $\hat{H}$ given by Eq. (B17). The storage modes are initialized in a product state $| β_{1} ⟩ | β_{2} ⟩$ that does not lie in the code space but which is a steady state of Eq. (B22). Thus, when $Δ_{n} = 0$ (left plots), the evolution is (approximately) trivial. The left two plots thus also serve as Wigner plots of the initial state $| β_{1} ⟩ | β_{2} ⟩$ . However, when $| Δ_{1} - Δ_{2} | ≫ 4 | α |^{2} κ_{2}$ (right plots), the system evolves to the code space, defined here by $α = \sqrt{2}$ . (b) Validity of approximating Eq. (B22) by Eq. (B24). Master Eqs. (B22), (B24) are simulated (with decoherence added to each mode via the dissipators $κ_{1} D [\hat{a}]$ and $κ_{1} D [{\hat{a}}^{†} \hat{a}]$ ), and the expectation value of $1 - {\hat{P}}_{c}$ is computed once the system reaches its steady state. Here ${\hat{P}}_{c}$ denotes the projector onto the cat-code space, and the subscripts “actual” and “ideal” denote expectation with respect to the steady states of Eqs. (B22) and (B24), respectively. The ratio of expectations, plotted on the vertical axis, quantifies the relative increase in population outside the code space. A ratio approximately $1$ indicates the approximation works well. Parameters are chosen from the ranges $| α |^{2} \in [1, 4]$ and $| Δ_{1} - Δ_{2} | / κ_{2} \in [5, 100]$ .
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Figure 23
Suppression of type I errors. (a) Plots of $κ_{eff} (M)$ as a function of the detuning, $δ$ , of the unwanted term. (b) Master-equation simulations. The system is initialized with a single excitation in the storage mode and evolved according to the dynamics $\dot{\hat{ρ}} = - i [(g_{2} \hat{a} {\hat{b}}^{†} e^{i δ t} + H . c .) + {\hat{H}}_{buffer + filter}, \hat{ρ}] + D [{\hat{L}}^{(3)}] (\hat{ρ})$ . These dynamics are analogous to those generated by ${\hat{H}}^{(3)}$ and ${\hat{L}}^{(3)}$ ; in both cases the unwanted term induces losses at rates $κ_{eff} (M)$ . Simulation results are indicated by open circles, and the analytical expressions for $κ_{eff} (M)$ are plotted as solid lines. Parameters: $α = \sqrt{2}, κ_{c} / g_{2} = 10, J / g_{2} = 5.$ For (b), $δ = 4 J$ , as indicated by the dashed line in (a).
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Figure 24
Suppression of type II errors. (a) Plots of $γ_{eff} (M)$ as a function of the detuning, $δ_{1} - δ_{2}$ , of the effective Hamiltonian. (b) Master-equation simulations. The storage mode is initialized in the even parity cat state and evolved according to the dynamics $\dot{\hat{ρ}} = - i [{\hat{\bar{H}}}^{(4)}, \hat{ρ}] + D [{\hat{L}}^{(4)}] (\hat{ρ})$ . Simulation results are indicated by open circles, and the analytical expressions for $γ_{eff} (M)$ are plotted as solid lines. Parameters: $α = \sqrt{2}, κ_{c} / g_{2} = 10, J / g_{2} = 5$ . Rather than specify values for $g$ and $δ_{1, 2}$ , we simply fix $χ_{eff} (M) / g_{2} = 0.2$ . For (b), $δ = 3 J$ , as indicated by the dashed line in (a).
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Figure 25
Cat-qubit stabilization in the surface-code architecture. Each ATS is coupled to two data modes $α, γ$ and two ancilla modes $β, δ$ . In practice, ATSs are also coupled to a fifth readout mode (not shown here because it is not stabilized by any ATS). Each ATS is responsible for performing four cnot gates (at different time steps) and stabilizing two phononic modes in the cat-code manifold during each time step. In the top panel, we show configurations of the cat-qubit stabilization, which respect the constraint discussed in the main text: at each time step, a cnot’s target mode must be stabilized by an ATS that also couples to its control mode. Each phononic mode, pointed by a black arrow, is stabilized by an ATS where the black arrow originates from. In the bottom panel, we show two stabilization configurations in the perspective of each host ATS. In configuration $1$ ( $2$ ), modes $α, β$ ( $γ, δ$ ) are stabilized by the host ATS and the remaining modes $γ, δ$ ( $α, β$ ) are stabilized by some other neighboring ATSs.
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Figure 26
Type III crosstalk errors in the surface-code architecture. We define $p_{double}$ as the probability of getting a type III error $\propto {\hat{Z}}_{α} {\hat{Z}}_{γ} {\hat{I}}_{β}$ , and $p_{triple}$ as the probability of getting a type III error $\propto {\hat{Z}}_{α} {\hat{Z}}_{γ} {\hat{Z}}_{β}$ .
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Figure 27
Optimized mode frequencies. (a) Plot of the optimized frequencies of the five storage modes. (b) Emitted buffer-mode photon detunings. Red dashed (solid) lines indicate photons emitted via parity-non-preserving type I (type II) processes. The yellow box covers the region $[- 50, 50] (2 π \times MHz)$ , representing a bandpass filter with center frequency $ω_{b}$ and a $4 J = 2 π \times 100$ MHz passband. The fact that no lines lie inside the yellow box indicates that all type I and II processes are sufficiently far detuned so as to be suppressed by the filter. The top (bottom) plot in (b) is for the case where modes $α$ and $β$ ( $γ$ and $δ$ ) are stabilized simultaneously.
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Figure 28
Log-log plot of Pauli $Z$ -type error rates for the cnot gate at optimal gate time with mean phonon number $n = 8$ in the presence of pure loss at rate $κ_{1}$ . The fits are performed over the range $κ_{1} / κ_{2}$ from $10^{- 4}$ to $10^{- 5}$ . The error rates $Z_{2}$ and $Z_{1} Z_{2}$ differ by no more than $10^{- 5}$ .
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Figure 29
Plot of Pauli $Z$ -type error rates for the cnot gate with mean phonon number $n = 10$ at various values of the gate time. The noise model is at rate $κ_{1} = 10^{- 5} κ_{2}$ , dephasing at a rate $κ_{ϕ} = κ_{1}$ , and gain with $n_{th} = 1 / 100$ . The gate time is plotted relative to the optimal gate time for these parameters. The optimal gate time minimizes the total error. The dotted curves are a linear fit for the $Z_{2}$ -error rate and a sum of a linear term and a $1 / T$ term for the $Z_{1}$ and total error rates, representing the contributions of loss and nonadiabatic errors.
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Figure 30
Plot of the fit parameters of the square root fit as shown in Fig. 28 for the Pauli $Z$ -type error rates of the cnot gate for different values of mean phonon number $n$ . The noise model in this figure is pure phonon loss.
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Figure 31
Plot of identity minus the super operator for the cnot noise channel in the Pauli basis. The cnot parameters are $n = 4$ , $κ_{1} / κ_{2} = 10^{- 5}$ , $n_{th} = 1 / 100$ , and $κ_{ϕ} = κ_{1}$ . The diagonal components of the matrix are the Pauli infidelities, in other words, one minus the probabilities that the cnot noise channel maps a given Pauli operator back to itself. The off-diagonal components represent the coherent part of the noise channel. These terms are orders of magnitude smaller than the dominant noise terms. The dominant $Z_{1}$ -error rate manifests itself as the relatively larger diagonal terms that are sensitive to a $Z_{1}$ error, i.e., Pauli operators with $X$ or $Y$ on the first qubit.
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Figure 32
Log-log plot of the Pauli error rates for the cnot gate with parameters, $n = 4$ , $κ_{1} = κ_{ϕ}$ , and $n_{th} = 1 / 100$ . Each of these error rates scale like $\sqrt{κ_{1} / κ_{2}}$ .
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Figure 33
Log-log plot of the smallest error rates for the cnot gate with parameters, $n = 4$ , $κ_{1} = κ_{ϕ}$ , and $n_{th} = 1 / 100$ . These error rates are proportional to $κ_{1} / κ_{2}$ rather than the square root scaling of the other error rates in Fig. 32.
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Figure 34
Plot of the Pauli error rates at a fixed value of the noise parameters and different values of the shifted Fock-basis dimension $d$ . Each error rate is scaled by its value at largest value of dimension $d = 9$ to show the convergence as $d$ increases. The parameters are set to $n = 3$ , $κ_{1} = 10^{- 5}$ , $κ_{ϕ} = 0$ , and $n_{th} = 0$ . One set of Pauli error rates converges rapidly as $d$ increases. This includes all Pauli errors where $Z$ or $I d$ act on the control qubit. Another set of Pauli errors with $X$ or $Y$ acting on the control qubit require much higher Hilbert-space dimension to capture accurately. This implies that these error rates include significant contributions from highly excited states.
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Figure 35
Log-linear plot showing exponential decay of non- $Z$ -error rates as mean phonon number $n = | α |^{2}$ increases. Four values of dephasing are plotted, $κ_{ϕ} = 0$ , $1$ , $2.5$ , and $10$ times $κ_{1}$ . The asterisk that appears for the nonzero values of dephasing represents that these points include phonon gain at a rate $n_{th} = 1 / 100$ . No gain is present in the $κ_{ϕ} = 0$ points. For each set of noise parameters, the upper set of points represents the Pauli error rates from Fig. 32 that scale with $\sqrt{κ_{1} / κ_{2}}$ and exponentially with $n$ . The lower set of points are the Pauli error rates from Fig. 32 that scale linearly with $κ_{1} / κ_{2}$ and exponentially with $n$ . The parameters of the exponential fits can be found in Table 2.
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Figure 36
Log-log plot of the various $Z$ -type error rates for the Toffoli gate at optimal gate time with parameters $n = 8$ , $κ_{ϕ} = κ_{1}$ , and $n_{th} = 1 / 100$ . These error rates were obtained in a shifted Fock-basis simulation using $d = 4$ for a total Hilbert-space dimension of 8 for each of the three cavities involved in the Toffoli gate. Qubits 1 and 2 are the controls and 3 is the target.
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Figure 37
Plot of the $Z$ -error rates for the $Z$ and CZ gates as a function of loss for $n = 10$ . The noise model for this plot is pure phonon loss with no gain. Gain will have a small effect of these error rates, while dephasing noise will have only a negligible effect. The dotted curves are best fits in the form $c * \sqrt{κ_{1} / κ_{2}}$ .
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Figure 38
Plot of the $Z$ -error rate parameters from best fits like the ones in Fig. 37 as a function of mean phonon number $n$ . The error rates are the product of these fit parameters and the loss rate $\sqrt{κ_{1} / κ_{2}}$ . The dotted best-fit curves in this plot are fits to $c / α$ where $α = \sqrt{n}$ . For small values of $α$ the scaling differs somewhat from the $1 / α$ scaling in the large $α$ limit. For this reason the fits were performed over the range $n = 6$ to $10$ .
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Figure 39
Circuit used for an $X$ -basis measurement in the context of an $X$ -type stabilizer measurement. The first step consists of entangling the ancilla qubit with the data qubits. Afterwords, the ancilla qubit is deflated followed by a swap with a readout mode. Lastly, the readout mode is repeatedly measured using a transmon qubit. The durations for the parts of the measurement procedure are labeled at the bottom of the figure below each circuit element. While these repeated parity measurements are occurring, the cnot gates of the next error-correction cycle can begin. Also included is a diagram of the physical layout of the stabilizer to give context to the measurement circuit.
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Figure 40
Simulated (markers) and analytical (curves) error probabilities for $X$ measurement for the cases of both even parity and odd parity initial states. We have taken the parity measurement to be QND and projective. The dependence on $κ_{1} / κ_{2}$ is stronger for the case of an odd initial state since the cavity is mapped to $| 1 ⟩$ after the deflation. The plotted curve is the leading-order analytic model for the case of $κ_{ϕ} = 0$ Eq. (G4). Numerical imprecision predominantly due to the deflation simulation can have about a 10 percent effect on the simulated infidelities for the smaller $κ_{1} / κ_{2}$ .
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Figure 41
(a) Measurement error probability for $Z$ -basis readout as a function of time. The colored lines correspond to the analytic formula for the separation error Eq. (G16) for $α^{2} = 1, 2, \dots, 10$ . The black lines correspond to infidelities from simulations of the corresponding stochastic master equation Eq. (G17) (QuTiP) in the interaction picture for a few thousand trajectories with the initial condition $| α ⟩$ for the cases $α^{2} = 1, α^{2} = 3, α^{2} = 5$ , and $α^{2} = 7$ . The simulated curves, which include $κ_{1} / 2 π = 1 KHz$ , and analytic curves agree well indicating the small effect of the additional single-phonon loss. In order to get concrete time numbers, the simulations use $1 / κ_{2} = 100$ ns but as discussed this can be scaled to any $κ_{2}$ . (b) Plot of the minimum infidelities versus $α^{2}$ and the fit line.
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Figure 42
Distance-5 repetition code, where one round of stabilizer measurements is performed (green measurements) followed by a direct measurement of the data qubits (red measurements). The data qubits are the yellow rectangles, and the ancilla qubits (prepared in $| + ⟩$ ) are the gray rectangles. On the left, a measurement error on the third data qubit occurs during the direct measurement of the data, which is equivalent to having a $Z$ data qubit error immediately before the measurement (shown on the right). Such settings illustrate the importance of the STOP algorithm, where one might have to correct errors prior to applying a non-Clifford gate, and a round of perfect error correction (which in practice is achieved by directly measuring the data) cannot be performed. In such settings, a single measurement error during the last round of stabilizer measurements (green measurements in the figure) can lead to a logical failure if the syndrome measurement is repeated a fixed number of times (say $d$ ) rather than using the STOP algorithm.
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Figure 43
Alternative architecture. The $X$ -basis readout is performed directly using the ATS [contained within the reservoir (RESVR)], as opposed to using an ancillary readout mode and transmon qubit, cf. Fig. 2. By eliminating the readout mode, the number of modes per unit cell is reduced from five to four.
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Figure 44
Logical $Z$ -error rates of the thin rotated surface code with five modes coupled to an ATS (which includes a transmon qubit and an additional readout mode in each unit cell and corresponds to the data in Fig. 8) and four modes coupled to an ATS (which excludes the transmon qubit and performs direct $X$ -basis measurements). We labeled measurements with the transmon qubit as low measurement and the direct $X$ -basis measurement as high measurement. For the direct $X$ -basis measurement, the measurement error rate is fixed at $2 \times 10^{- 3}$ for all values of $κ_{1} / κ_{2}$ . Measurement error rates with the transmon qubit were obtained from Table 3 with five parity measurements.
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Figure 45
Logical $Z$ failure rates for a $d_{x} = 3$ and $d_{z} = 11$ thin surface code in the presence of the residual crosstalk errors (given in Table 10). The $X$ -basis measurement error rates are obtained from Table 3 with five parity measurements. We compute the logical $Z$ -error rates for different values of $g_{2}$ shown in the legend, and compare such results to the case where the crosstalk errors are not present. The results in (a) [respectively, (b)] are obtained with crosstalk error rates taken from the five-mode (four-mode) row of Table 10 with $| α |^{2} = 8$ .
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Figure 46
Example of a single controlled- $Z$ failure resulting in the error $Z_{n} \otimes Z_{n + 1}$ (where $Z_{n + 1}$ acts on the ancilla qubit) when measuring the operator $Z^{\otimes n}$ . Here $n$ is the number of data qubits. This single fault can cause three consecutive syndrome measurements to yield three distinct outcomes. Here $E_{in}$ is an input error with syndrome $s (E_{in}) = s_{1}$ .
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Figure 47
Circuit for preparing logical computational basis states of the repetition code. A measurement error results in a logical $X$ error applied to the data. Fault tolerance is achieved by repeating the measurement using the STOP algorithm.
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Figure 48
Circuit for implementing a logical $S$ gate. The circuit requires the preparation of $| i ⟩_{L}$ , and the cnot gate is transversal. A logical $| Z ⟩_{L}$ operator is applied when the measurement outcome of the ancilla is $- 1$ .
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Figure 49
Circuit for implementing a logical $Q = S H S$ gate. The circuit requires the preparation of $| i ⟩_{L}$ , and the cnot gate is transversal. A logical $| Y ⟩_{L}$ operator is applied when the measurement outcome of the ancilla is $- 1$ .
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Figure 50
Efficient circuit for implementing a $C Z$ gate given the higher cost of logical $H$ gates compared to logical $S$ gates.
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Figure 51
Diagram illustrating our protocol for growing the state $| ψ ⟩_{d_{1}}$ to $| ψ ⟩_{d_{2}}$ with the ATS layout by starting with the two blocks stabilized by $S_{d_{1}}$ and $S_{d_{1}^{'}}$ . The yellow vertices are the data qubits, and the gray vertices correspond to the ancilla qubits used to measure the stabilizers of the repetition code. The measurement of $X_{d_{1}} X_{d_{1} + 1}$ (with random $\pm 1$ outcome) is highlighted by the purple semicircle. After performing the appropriate corrections, the final block is stabilized by $S_{d_{2}}$ .
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Figure 52
The three stages of lattice surgery corresponding to cross sections (time slices) of the lattice-surgery space-time diagram in Fig. 11. Step 1 prepare: data qubits between the surface-code blocks are prepared in the $| 0 ⟩$ state. Step 2 merge: start measuring the $Z$ and $X$ stabilizers indicated. The product of the $X$ stabilizers (highlighted with white vertices) yields the outcome $X_{L 1} X_{L 2}$ . However, a measurement error on a white vertex will flip the outcome and so these stabilizer measurements must be repeated $d_{m}$ times, with $d_{m}$ chosen sufficiently large to suppress timelike errors to the desired probability. Step 3 split: The qubits between the initial surface-code blocks are measured in the $Z$ basis. Note that it is not possible to use $X$ -basis measurements to disentangle as this would measure $X_{L 1}$ and $X_{L 2}$ . If the parity of the single-qubit $Z$ measurements is “ $- 1$ ” then we must apply a Pauli correction $X_{L 1}$ (or equivalently $X_{L 2}$ ) as a correction. Both the measurement of $X_{L 1} X_{L 2}$ and the estimated Pauli correction must be done fault tolerantly after having decoded the syndrome. In the case of $X_{L 1} X_{L 2}$ , we choose one particular time slice $t_{p}$ and make an initial guess by multiplying all the white vertices at time $t_{p}$ . If the decoder assigns a measurement error to any white vertex at time $t_{p}$ , then we must account by flipping the $X_{L 1} X_{L 2}$ . If the accumulated physical $Z$ errors before time $t_{p}$ anticommute with $X_{L 1} X_{L 2}$ then we flip the outcome. For a similar discussion of lattice surgery see Ref. [124]. Compared to Figs. 2 and 5 we use a similar graphical representation but for simplicity omit the location of the transmon, readout qubit, and ATS.
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Figure 53
Fitted results for simulation of $d_{x} = 3$ surface code for logical $Z$ and timelike errors. We fit according to the ansatz of Eqs. (M2) and (M3). (Top) The logical $Z$ simulations for which we set $t = d_{z}$ and plot the error probability divided by $t$ . (Bottom) the probability of a timelike error during lattice surgery for which we set $ℓ = d_{m} - 1$ . All data points shown are used in fitting. This is a truncated data set eliminating points above $10^{- 4}$ on the error rate axis and eliminating points outside the relevant range of $κ_{1} / κ_{2}$ .
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Figure 54
(a) Circuit for measuring the stabilizers of the $d = 5$ repetition code. The dark rectangular boxes correspond to idling qubit locations. (b) MWPM decoding graph for the $d = 5$ repetition code where the syndrome measurement is repeated 5 times.
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Figure 55
(a) Surface-code lattice with $d_{x} = 5$ and $d_{z} = 7$ . (b) Graph used for decoding $X$ stabilizer measurement outcomes with both bulk and boundary edge-weight probability labels. (c) Graph used for decoding $Z$ -stabilizer measurement outcomes with both bulk and boundary edge-weight probability labels.
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Figure 56
(a) Graph used for decoding $X$ -type stabilizer measurements, which include vertical edges (dashed gray edges) for dealing with measurement errors and space-time correlated edges for correcting errors arising from cnot gate failures causing two different syndrome measurement outcomes in consecutive rounds. (b) Same as in (a) but for $Z$ -type stabilizer measurements.
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Figure 57
(a) Fictitious identity gates illustrating the possible correlated errors arising before the $X$ -basis measurement of the $X$ -type ancilla qubits. Gray squares correspond to the first qubit, blue triangles to the second qubit and green circles to the third qubit. (b) $X$ -type decoding graph with added edges to correct correlated errors. The edge-weight probabilities of the orange cross edges are labeled $P_{cross}$ . We also add red edges with edge-weight probabilities labeled $P_{d, corr}$ at the bottom row of the graph.
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Figure 58
Example of a decoding graph for correcting timelike errors using a $d = 5$ repetition code with $d_{m} = 4$ . The top and bottom boundary edges (with zero weight) and vertices are shown in blue and are connected by a blue edge with zero weight. As explained in Appendix pp13, we have removed the left and two-dimensional black edges (which correspond to the left and rightmost qubits) to isolate timelike errors.
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Figure 59
(a) Implementation of the timelike decoding protocol in the presence of a single measurement error when measuring the stabilizer $X_{2} X_{3}$ during the first round. The minimum weight path matches to the bottom boundary going through the vertex $v_{2}^{(1)}$ whose outcome is correctly flipped (illustrated by the yellow star). (b) Same as in (a) but with an additional measurement error occurring in the second round when measuring $X_{2} X_{3}$ . In this case, the minimum weight path matches to the top boundary and fails to flip the measurement outcome of $v_{2}^{(1)}$ (which is incorrect given the measurement error in the first round) resulting in a logical failure.
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Figure 60
A magic state distillation protocol for $8 CCZ \to 2 CCZ$ with the eight $CCZ$ injections performed using Algorithm 64. Qubit labels of form $(D, i)_{f}$ and $(D, j)_{BU}$ follow notation of Definition 2. For each $j$ , the triple of qubits $(A, 1)_{BU}$ , $(B, 2)_{BU}$ , and $(C, 3)_{BU}$ are prepared in a noisy $CCZ$ states (e.g., using BUTOF) but this preparation is not shown. We show explicitly the cnot gates for the first two steps and the last step, but omit the middle steps for brevity. The full circuit is reproducible using Eqs. (O24) to (O26) to specify the cnot pattern as outlined in Algorithm 64.
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Figure 61
Lattice surgery to measure a multipatch Pauli measurement between three code blocks: one repetition-code logical qubit and two thin surface-code logical qubits. See Fig. 52 for comparison. This provides the principle building block for the execution of item 1(a). In TDTOF, a multipatch Pauli measurement is always followed by a single-qubit measurement of the repetition code and here we combine this with the third (split) step of lattice surgery. The gradient colored squares represent where an $X \otimes X \otimes Z \otimes Z$ stabilizer measurement called a domain wall. Multiplying the outcome of the stabilizers labeled with a white dot, gives the outcome of the $Z_{L} \otimes Z_{L} \otimes Z_{L}$ multipatch Pauli measurement [this is how we obtain the outcomes labeled $ω_{j}^{D}$ in item 1(a) of Algorithm 65]. To ensure fault tolerance of this measurement outcome, we repeat these stabilizer measurements $d_{m}$ times as discussed in Appendix 40. Afterwards all qubits are measured in $Z$ basis, with their product determining the operator $Z_{L} \otimes 1 \otimes 1$ [this is how we obtain the outcomes labeled $m_{j}^{D}$ in item 1(a) of Algorithm 65]. Since an error during a $Z$ measurement is physically indistinguishable from a bit-flip error prior to the measurement, to fully exploit noise bias we need these failures to be similarly unlikely. However, in practice, in our noise model we find $Z$ measurement errors are much more likely than bit flips. However, since the qubits are idle for a long time after the $Z$ measurement, we can boost their effective fidelity by using an ancilla qubit to perform a nondestructive $Z$ measurement and repeat until the fidelity is suitably high. As such, here the single-qubit $Z$ measurements should be interrupted as repeated nondestructive $Z$ measurements. We discuss in Appendix 45 the effect of errors in lattice surgery due to using finite-size code blocks. If we wish to instead measure $X_{L} \otimes Z_{L} \otimes Z_{L}$ then we do not use the domain wall on the repetition-code block.
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Figure 62
A 2D layout for realizing $8 TOF \to 2 TOF$ distillation via lattice surgery using a mixture of repetition codes and thin surface code. Example dimensions shown here with encoding distances $d_{x} = 3$ , $d_{rep} = 5$ , $d_{z} = 5$ , and $D_{z} = 7$ ; and $M = 10$ BUTOF modules. Additional space between code blocks is provided for lattice surgery and routing between code blocks (see Appendix pp13 and Fig. 61). We give explicit locations for the nine factory qubits with labels $(D, i)_{f}$ following Definition 2. The modules labeled BU consists of three repetition codes and provide space to attempt a noisy $| TOF ⟩$ preparation using the BUTOF protocol. Note that BUTOF is executed with a distance $d_{BU}$ repetition code (typically we set $d_{BU} = 5, 7$ ) and then immediately grow to distance $d_{z} > d_{BU}$ .
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Figure 63
STOP algorithm
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Figure 64
A cnot circuit realizing $V^{†} {CCZ}^{\otimes n} V$ as defined in Appendix 40. Uses a trio of $G$ matrices with $n$ rows and $k + m$ columns. Qubit label convention given in Definition 2.
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Figure 65
A Pauli measurement scheme realizing $V^{†} {CCZ}^{\otimes n} V$ as defined in Appendix 40. Uses a trio of $G^{D}$ matrices with $n$ rows and $k + m$ columns. Qubit label convention given in Definition 2.
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Figure 66
A complete magic state distillation routine using a space-time trade-off and multiqubit Pauli measurements. It assumes a trio of $G^{D}$ matrices of size $n \times (m + k)$ representing $[[n, k, d]]$ codes with $CCZ$ transversality as in Lemma 2. We gave suitable $G^{D}$ matrices in Eqs. (O24) to (O26) for which we have $n = 8$ , $k = 2$ , $d = 2$ , and $m = 1$ . Qubit label convention given in Definition 2
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PRX Quantum

a Physical Review journal

Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes

Christopher Chamberland, Kyungjoo Noh, Patricio Arrangoiz-Arriola, Earl T. Campbell, Connor T. Hann, Joseph Iverson, Harald Putterman, Thomas C. Bohdanowicz, Steven T. Flammia, Andrew Keller, Gil Refael, John Preskill, Liang Jiang, Amir H. Safavi-Naeini, Oskar Painter, and Fernando G.S.L. Brandão

PRX Quantum 3, 010329 – Published 23 February 2022

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