Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes

Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes Daniel Litinski and Felix von Oppen Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany We present a planar surface-code-based bilizers requires more potentially faulty controlled-not scheme for fault-tolerant quantum computation (CNOT) gates. which eliminates the time overhead of single- The main drawback of surface codes in comparison qubit Clifford gates, and implements long-range to color codes is the absence of transversal single-qubit multi-target CNOT gates with a time overhead Clifford gates, i.e., the gates that are products of the that scales only logarithmically with the control- Hadamard gate H and the phase gate S. While the target separation. This is done by replacing transversal Clifford gates of color codes provide them hardware operations for single-qubit Clifford with fast logical H and S gates, defect-based propos- gates with a classical tracking protocol. Inter- als for surface codes [14] implement the H gate via a qubit communication is added via a modified multi-step measurement protocol, and the S gate via a lattice surgery protocol that employs twist de- distilled ancilla qubit. In order to lower the overhead fects of the surface code. The long-range multi- of single-qubit Clifford gates, surface code qubits can target CNOT gates facilitate magic state distil- be encoded using twist defects [15], which are essen- lation, which renders our scheme fault-tolerant tially Majoranas that can be braided via code defor- and universal. mation [16]. It was pointed out that braiding of twists can also be implemented via a classical tracking protocol [17], in accordance with the Gottesman-Knill theo- 1 Introduction rem [18]. In this work, we present a scheme that implements The performance of quantum computers is limited by this tracking protocol for planar surface codes, as op- the coherence times of the underlying physical qubits. posed to twist-based encodings. We refer to this pro- Quantum error correction [1] offers the possibility to tocol as edge tracking. In our scheme, Clifford com- enhance the qubits' survival times by encoding quan- pleteness is achieved via a modified lattice surgery [19] tum information using logical qubits consisting of many physical qubits. Topological quantum error-correcting codes [2,3] are of particular interest, as they only require the measurement of spatially local operators { Stabilizers: a feature that is compatible with the local operations accessible in two-dimensional solid-state qubit ar- Z Z Z d chitectures, such as superconducting qubits [4], spin ⊗ qubits [5], or Majorana-based qubits [6]. Z = Z Z Z Quantum error-correcting codes typically operate in edge arXiv:1709.02318v2 [quant-ph] 17 Apr 2018 cycles. In each code cycle, mutually commuting oper- X X X ators called stabilizers [7] are measured to reveal the Z error syndrome, which is used to determine and cor- X X X rect errors. Surface codes [8,9] are topological codes that feature a high error threshold [10, 11], and only re- X = X⊗d quire the measurement of four-qubit stabilizer operators edge for the readout of the error syndrome. The low-weight Figure 1: An example of a surface code qubit with code distance stabilizers are an advantage over other codes such as d = 5. Physical qubits are located on the vertices, and the color codes [12, 13], which require the measurement of faces define the two- and four-qubit Z type (bright) and X six-qubit operators. This facilitates syndrome readout type (dark) stabilizer operators. X strings along the X edge in many physical architectures such as superconducting (orange) are logical XL operators, whereas Z strings along Z qubits, where the measurement of higher-weight sta- edges (blue) are ZL operators. Accepted in Quantum 2018-16-04, click title to verify 1 H-gate S-gate XL ZL ZL XL ZL YL Figure 2: An example of edge tracking with a wide surface code qubit. Starting from the default encoding Xedge = XL and Zedge = ZL, an H gate changes it to Xedge = ZL and Zedge = XL, and a subsequent S gate modifies it to Xedge = ZL and Zedge = YL. protocol. Twist defects are no longer used to encode code distances. quantum information, but reappear in lattice-surgery Surface code qubits have two distinct types of bound- protocols involving the logical YL operator, so that we aries, usually referred to as rough and smooth edges. refer to the protocol as twist-based lattice surgery. Our Here, we call them X and Z edges in analogy to the log- scheme provides long-range multi-target CNOT gates { ical Pauli operators XL and ZL that they encode. Sur- i.e., CNOTs with one control and arbitrarily many tar- face code qubits can be easily initialized in the logical gets { between any set of edge-tracked surface code +1-eigenstates |0Li and |+Li of ZL and XL by initial- qubits. These gates are particularly useful for magic izing all physical qubits in the corresponding physical state distillation [20], which completes the universal states |0i and |+i, measuring all stabilizers, and correct- gate set by fault-tolerantly implementing the T gate ing the errors. Conversely, they can be read out in the (or π/8 gate). Our scheme not only eliminates the need XL and ZL basis by measuring all physical qubits in the for hardware operations for single-qubit Clifford gates, X or Z basis, and performing classical error correction. but also conceptually simplifies the twist-defect-based We define the operator X (Z ) as the string approach to surface-code quantum computing. Even edge edge of X operators (Z operators) on all physical qubits though our scheme features twist defects and disloca- along an X edge (Z edge). In the default encoding, tion lines, the only concepts necessary to understand X = X and Z = Z . The edge tracking proce- our scheme are the encoding of logical qubits and the edge L edge L dure that we now introduce essentially modifies which measurement of two-qubit parity operators. We discuss logical operators are encoded by X and Z . Logi- the implementation of the single-qubit Clifford gates, edge edge cal single-qubit Clifford gates map the logical Pauli op- CNOT gates, and T gates in Secs.2,3 and4, respec- erators X , Y , and Z onto other Pauli operators. In tively. In a concluding section, we discuss our scheme in L L L particular, an H gate maps X → Z , Y → −Y , and the context of possible hardware implementations and L L L L Z → X . An S gate maps X → Y , Y → −X , and in comparison to alternative topological codes. L L L L L L ZL → ZL. Thus, we can replace single-qubit Clifford gates by a classical tracking procedure. This is essen- 2 Edge Tracking tially the content of the Gottesman-Knill theorem [18], which states that Clifford gates can be simulated effi- The basic framework of our scheme are physical qubits ciently on a classical computer. For now, we only con- arranged on a 2D square lattice which allow for the sider tracking of single-qubit Clifford gates H and S, measurement of local stabilizer operators. Examples of whereas CNOT gates are performed explicitly. possible physical realizations include superconducting In order to combine this tracking scheme with lat- qubits emulating stabilizer measurements using ancilla tice surgery, it will be convenient to use the wide qubits and CNOT gates [14], or Majorana-based qubits qubits shown in Fig.2 instead of the square qubits using direct measurements of the stabilizers via Majo- that were previously introduced. These qubits have rana fermion parity measurements [21]. A single surface an X and Z edge on the same side, such that the code qubit can be defined using the checkered square logical operators XL, YL and ZL can all be accessed shown in Fig.1, where physical qubits are located at by lattice surgery from the same side of the qubit. the vertices. We refer to the Pauli operators of the Compared to square qubits with the same code dis- physical qubits as X, Y , and Z. The faces define the tance, this comes at the price of a larger number X⊗n- and Z⊗n-stabilizers of the code, where n is the of physical qubits for each logical qubit. The fig- number of qubits that are part of the face. The figure ure also shows an example of edge tracking. The de- shows an example of a code with code distance d = 5, fault encoding is Xedge = XL and Zedge = ZL. An but this construction can be generalized to arbitrary H gate changes the encoding to Xedge = ZL and Accepted in Quantum 2018-16-04, click title to verify 2 Zedge = XL. A subsequent S gate modifies it to col. The measurement outcome of the orange stabiliz- Xedge = ZL and Zedge = YL. ers is trivial, as they are products of previously known boundary stabilizers. The outcome of the blue stabilizers, on the other hand, is nontrivial. They contain each 3 Lattice surgery with a twist boundary qubit exactly once. Therefore, their product is precisely the operator Z(control) ⊗Z(ancilla), which cor- Edge tracking requires a suitable CNOT gate protocol edge edge responds to the Z ⊗ Z parity in the default encoding. in order to be useful for universal quantum computa- L L Thus, lattice surgery implements a fault-tolerant par- tion. This is provided by twist-based lattice surgery. ity measurement between logical qubits. Similarly, in It essentially implements the circuit identity shown in the following lattice surgery step (3), the blue stabiliz- Fig.3 for edge-tracked qubits.

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