Fault-tolerant fermionic quantum computation based on color code

Fault-tolerant fermionic quantum computation based on color code

Ying Li Graduate School of China Academy of Engineering Physics, Beijing 100193, China Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom
July 5, 2019
Abstract

An important approach to the fault-tolerant quantum computation is protecting the logical information using the quantum error correction. Usually, the logical information is in the form of logical qubits, which are encoded in physical qubits using quantum error correction codes. Compared with the qubit quantum computation, the fermionic quantum computation has advantages in quantum simulations of fermionic systems, e.g. molecules. In this paper, we show that the fermionic quantum computation can be universal and fault-tolerant if we encode logical Majorana fermions in physical Majorana fermions. We take a color code as an example to demonstrate the universal set of fault-tolerant operations on logical Majorana fermions, and we numerically find that the fault-tolerance threshold is about .

I Introduction

Quantum computation can solve some problems much faster than classical computation Nielsen2010 (), e.g. simulating large quantum systems. Because quantum states are fragile, fault tolerance is crucial to quantum computation. There are two approaches to the fault-tolerant quantum computation: error-correction-based quantum computation Shor1996 () and topological quantum computation Nayak2008 (). In the error-correction-based quantum computation, the quantum information is encoded using quantum error correction codes and protected by actively detecting and correcting errors. In the topological quantum computation, the quantum information is encoded in the state of non-Abelian anyons and processed by braiding anyons, and these braiding operations can tolerate small perturbations.

Majorana fermions are non-Abelian anyons considered as candidates for the topological quantum computation Nayak2008 (), which have been observed in recent experiments Mourik2012 (); Mebrahtu2013 (); NadjPerge2014 (); Xu2015 (); Haim2015 (); Albrecht2016 (); Deng2016 (); He2017 (). Majorana fermions can also be used for implementing the fermionic quantum computation, in which fermionic modes instead of qubits are the basic units that carry the quantum information Bravyi2002 (). The fermionic quantum computation is polynomially equivalent to the qubit quantum computation but has advantages in quantum simulations of fermionic systems, e.g. molecules Wecker2014 (); Poulin2015 (); Reiher2016 (); Kassal2011 (). By encoding each qubit in four Majorana fermion modes Bravyi2006 (), the fermionic quantum computation can efficiently simulate the qubit quantum computation. However, to simulate fermions using qubits, fermions need to be encoded in the state of the entire qubit system as a whole, and some local operations on fermions are realised by qubit operations on a significant portion of total qubits Bravyi2002 (); Ortiz2001 (); Whitfield2011 (); Jones2012 (); Hutter2015 (); Seeley2012 (); Tranter2015 (). These non-local qubit operations cost resources and may slow down the quantum computing.

Quantum computation based on Majorana fermions still needs the quantum error correction Bravyi2010 (); Terhal2012 (); Vijay2015 (); Landau2016 (); mypaper (); Plugge2016 (); Vijay2017 (); Litinski2017 (). Braiding Majorana fermions does not provide all the operations required by the universal quantum computation, therefore some topologically unprotected operations have to be introduced to complete the operation set Bravyi2006 (). These unprotected operations could cause errors, which need to be corrected using the quantum error correction. Majorana fermions may also suffer change-tunnelling errors caused by the fermionic bath in the environment, e.g. unpaired electrons in the superconductor Goldstein2011 (); Trauzettel2012 (); Cheng2012 (); Loss2012 (); Campbell2015 (). In order to correct errors, we can either encode physical qubits in Majorana fermions Bravyi2006 () and then encode logical qubits in physical qubits mypaper (); Plugge2016 (); Litinski2017 () or directly encode logical qubits in Majorana fermions Bravyi2010 (); Terhal2012 (); Vijay2015 (); Landau2016 (); Vijay2017 (). In either case, the logical machine is a qubit quantum computer. In this paper, we propose using logical Majorana fermions encoded in physical Majorana fermions to realise the genuine fermionic quantum computation. We demonstrate that the universal operation set can be implemented on logical Majorana fermions using local operations in a fault-tolerant manner.

Color code is a class of topological quantum error correction codes Bombin2006 (). We can construct codes for encoding logical Majorana fermions using color codes Bravyi2010 (). Taking a triangular color code as an example, we demonstrate the universal set of operations on logical Majorana fermions, and we also propose an efficient decoding algorithm for the code based on the code unfolding Bombin2012 (); Kubica2015 (). The error-rate threshold is an important measure of the code performance: only when the error rate is lower than the threshold, the fault-tolerant quantum computation can be realised. Usually, color codes provide much lower thresholds Landahl2011 () than high-threshold codes, e.g. the surface code. By numerically simulating the quantum error correction, we find that the threshold of the Majorana fermion color code is about for entangling operations, which is comparable to the threshold of the surface code Wang2011 ().

The paper is organised as follows. Codes for encoding logical Majorana fermions in physical Majorana fermions are introduced in Sec. II. We introduce the color code in Sec. III. The universal fault-tolerant fermionic quantum computation is discussed in Sec. IV, Sec. V and Sec. VI. The decoding algorithm is described in Sec. VII, and the numerical result about the fault-tolerance threshold is shown in Sec. VIII. A summary is given in Sec. IX.

Ii Majorana fermion codes for encoding logical Majorana fermions

Majorana fermions are described by Hermitian operators obeying , and each operator corresponds to a mode of Majorana fermions.

Majorana fermion codes are stabiliser codes of Majorana fermions Bravyi2010 (). Many quantum error correction codes, e.g. Calderbank-Shor-Steane codes, color codes Bombin2006 (), and the surface code Dennis2002 () are stabiliser codes of qubits Nielsen2010 (). For a code of physical Majorana fermion modes, the dimension of the Hilbert space is . The logical information is encoded in a subspace defined by the stabiliser group , in which each generator is a product of Majorana fermion operators , where is the set of modes supporting . We remark that changing the order of operators in the product may change the overall sign. The number of Majorana fermion modes in a stabiliser generator is always even, otherwise the generator changes the parity of the number of fermions, which is not physical. These stabiliser generators are Hermitian and mutually commutative. The phase makes sure that is Hermitian. The number of common modes for any two stabiliser generators is even, so that two generators and are commutative. Because , a stabiliser generator has two eigenvalues and . The logical subspace is the subspace that all stabiliser generators take the eigenvalue , i.e.  if the state is in the logical subspace. If there are independent stabiliser generators, the dimension of the logical subspace is , and there are total subspaces corresponding to sets of eigenvalues. In general, we can choose any of these subspaces as the logical subspace. Errors may bring the state out of the logical subspace, and they are detected by checking whether the state is still in the logical subspace and, if the state is not in the logical subspace, in which subspace the state ends up, i.e. measuring eigenvalues of stabiliser generators.

The logical state is described using logical operators. Logical operators are also products of Majorana fermion operators, and they commute with all stabiliser generators. For a logical operator defined on the set of modes , is always even for any stabiliser generator , so that . Here, is a phase. For example, we can encode one qubit in four Majorana fermion modes Bravyi2006 () , , and . The only stabiliser operator is , and logical operators are and , which are Pauli operators of the encoded qubit.

We can encode logical Majorana fermions in qubits. However, such an approach does not lead to the genuine fermionic quantum computation. Majorana fermions can be encoded using the Jordan-Wigner transformation and  Ortiz2001 (); Hutter2015 (), where the product is taken over all qubits with a number less than . Therefore, sometime we need to perform a sequence of qubit operations across the entire system to operate one Majorana fermion mode. For example, to perform the operation , we need to perform about gates on qubits between and  Whitfield2011 (). Although these gates can be implemented in a constant time by using ancillary qubits Jones2012 (), any other operation on fermionic modes from to cannot be performed in parallel with , because corresponding qubits are in use. In the worst case, the entire computer is occupied to operate only two fermionic modes, e.g. , where is the total number of fermionic modes. We remark that the locality of the encoding of individual fermionic modes could be improved by using the Bravyi-Kitaev transformation Bravyi2002 (); Seeley2012 (); Tranter2015 (). Fermionic modes can also emerge in qubit topological codes with twists Bombin2010 (); Brown2017 (), therefore, are protected by the code.

We can encode one logical Majorana fermion in an independent set of physical Majorana fermions Bravyi2010 (), so that each logical Majorana fermion can be operated independently to realise the genuine fermionic quantum computation. We consider a code of Majorana fermion modes with independent stabiliser generators, and the mode is not included in any stabiliser operator, i.e. . We remark that stabiliser generators should cover all other modes, so that any single-mode error can be detected. For such a code, is a logical operator. Because are all even, commutes with all stabiliser operators . This logical operator satisfies and . For any Majorana fermion operator other than these operators , we can find that the logical operator satisfies . If we consider two copies of the code, two logical operators and also satisfy . Therefore, the logical operator represents a logical Majorana fermion mode.

The mode is not included in neither stabiliser generators nor the logical operator, so it is redundant and can be removed, i.e. we can encode one logical Majorana fermion mode in physical Majorana fermion modes. We cannot define the Hilbert space of Majorana fermion modes, therefore we cannot define the logical subspace of the code directly. After introducing the -th mode, the Hilbert space of the total modes is -dimensional, and the logical subspace is -dimensional. It does not matter whether the -th mode is a physical Majorana fermion or a logical Majorana fermion. Therefore, for even number of logical Majorana fermion modes, we can define their common logical subspace.

Iii Color code

Figure 1: Triangular code based on the lattice. Each vertex represents a Majorana fermion mode, and each plaquette represents a stabiliser generator. The side length (the number of vertices on the side) of the lattice is .

Color code is a class of topological stabiliser codes Bombin2006 (). A color code is defined on a lattice, e.g. the lattice in Fig. 1, in which plaquettes can be colored with three colors (i.e. green, blue and red), and every two plaquettes sharing an edge have different colors. The number of vertices on any plaquette is always even, and the number of vertices shared by any two plaquettes is also always even. Therefore, we can construct stabiliser generators of Majorana fermions according to the color code lattice: a Majorana fermion mode is placed at each vertex, and the product of Majorana fermion operators on a plaquette is a stabiliser generator Bravyi2010 (). In other words, for each , the label denotes a plaquette, and denotes the set of vertices (i.e. Majorana fermion modes) on the plaquette. Because of properties of the lattice, these stabiliser generators satisfy that and are even. The code encodes one logical Majorana fermion mode if the total number of vertices is odd, e.g. triangular codes obtained by removing a vertex from a color code lattice defined on a sphere Bombin2006 () such as the code in Fig. 1. The logical operator is the product of all Majorana fermion operators.

In the following, we will take the color code in Fig. 1 as an example to discuss how to perform fault-tolerant fermionic quantum computation. This code is based on the lattice but different from the triangular code reported in Ref. Bombin2006 (). We are interested in this lattice, because we find that such a lattice provides a high error-rate threshold in the fault-tolerant qubit quantum computation based on Majorana fermions mypaper ().

Iv Universal fermionic quantum computation

Figure 2: Circuits for Majorana fermions. Each horizontal line represents a Majorana fermion mode. (a) A phase gate is equivalent to two exchange gates . (b) The state transfer operation transfers the state of to . Green boxes represent initialisation operations, and red boxes represent measurement operations. The phase gate depends on the measurement outcome of (indicated by the dotted line). (c) The exchange gate realised using a sequence of state transfer operations. (d) The gate realised using the magic state and state transfer operations. The magic state is the eigenstate with and , where . Outcome-dependent phases gates in and are equivalent to the outcome-dependent exchange gate (indicated by dotted lines), which can be realised using the circuit in (c). (e) The circuit for measuring the stabiliser operator of eight Majorana fermion modes using ancillary Majorana fermion modes . Each pair of ancillary modes is initialised in the eigenstate (). Parity projections are performed on each group of four Majorana fermion modes, and outcomes are . The measurement outcome of the stabiliser operator is . After parity projections, ancillary modes are measured, and outcomes are . To complete the stabiliser measurement, phase gates need to be performed according to these outcomes as shown in Table 1.

The operation set allowing universal fermionic quantum computation includes i) the initialisation of a pair of Majorana fermion modes in the eigenstate ; ii) the measurement of ; iii) the non-destructive measurement of , which is called parity projection in this paper; iv) the exchange gate , and v) the gate  Bravyi2002 (). Any parity-preserving unitary operator can be composed using these operations.

In this section, we show that the operation set is still universal if we replace the exchange gate with the phase gate and the gate with the preparation of the magic state. The magic state of the gate is an eigenstate of four Majorana fermion modes with and . Considering the code for encoding one qubit in four Majorana fermion modes (i.e. , and ), such an eigenstate can be written as , which is the magic state for the qubit non-Clifford gate  Bravyi2005 (). In the next two sections, we demonstrate how to implement the universal operation set, including the initialisation, measurement, parity projection, phase gate and magic-state preparation, on logical Majorana fermions.

We can realise the exchange gate using the phase gate provided that the initialisation and measurement are available, and vice versa. The exchange gate exchanges states of two Majorana fermion modes and adds a phase to one of them, i.e.  and . By exchanging two Majorana fermion modes twice, the phase is added to each of them, which is equivalent to a phase gate ( and ) [see Fig. 2(a)]. As we will show next, the state of a Majorana fermion mode can be transferred up to a random phase using the initialisation and measurement. Therefore, the exchange gate can be realised by firstly exchanging states of two Majorana fermion modes using the initialisation and measurement and then adjusting the phase using the phase gate.

The state transfer operation is described by the superoperator [see Fig. 2(b)]. Here, is the initialisation that prepares and in the eigenstate , and is the measurement of followed by a phase gate on and (i.e. ) depending on the measurement outcome . We note that is a projector to the eigenstate , and is a superoperator. Using , we get . Therefore, the state transfer operation is equivalent to an initialisation operation on and followed by an exchange gate on and , i.e. the state of is transferred to with an additional phase. The phase is determined because of the outcome-dependent phase gate. Here, we have assumed that the measurement is non-destructive. If the measurement is destructive, and may not be initialised in the correct eigenstate, but the sate of can still be transferred to with the correct phase.

The exchange gate can be realised using a sequence of state transfer operations [see Fig. 2(c)]. Such a combination of state transfer operations is equivalent to , i.e. an initialisation operation on and followed by an exchange gate on and . Here, we have used that .

The gate can also be realised using a sequence of state transfer operations. The overall operation is [see Fig. 2(c)], where . Initialisation operations and (in and , respectively) prepare the magic state on , , and , i.e. the eigenstate with and . Outcome-dependent phase gates on modes can be realised using phase gates and exchange gates as and . The overall operation is equivalent to , where we have used that . Because of the gate , the input magic state is consumed, i.e. the output state of , , and is the eigenstate with and . Therefore, the overall operation is also equivalent to , i.e. initialisation operations on , and , followed by a gate on and .

V Fault-tolerant operations and magic-state preparation

Figure 3: Operations on logical Majorana fermions. Each logical Majorana fermion mode is encoded in a triangular lattice, whose boundary is marked by the dashed line. Ancillary Majorana fermion modes are covered by white bars. Each white bar covers two vertices and denotes , where and are two corresponding Majorana fermion operators. At the beginning of the logical operation, ancillary modes are initialised according to white bars. Then stabiliser measurements are performed on the entire lattice to read eigenvalues of stabiliser operators. Stabiliser measurements are repeated for rounds. Finally, ancillary modes are measured according to white bars. In (a) and (c), the separation between dashed lines is . In (b), the thickness of the region above the dashed line is . In (d), the lattice is a square with the side length . Here, is the side length of logical Majorana fermions.

In this section, we discuss how to perform the phase gate, initialisation, measurement, parity projection, and preparation of magic states on logical Majorana fermions.

v.1 Phase gate

The phase gate is equivalent to two single-mode phase operations . The operation is not physical in a closed system, because it changes the parity of the number of fermions in the system, which is a conserved quantity in a closed system. We can realise by introducing ancillary Majorana fermion modes to play the role of the environment and then exchanging fermions between the system and environment. For this purpose, ancillary modes can be initialised in any state, and the overall input state is a product state . Here, is the state of Majorana fermions carrying the quantum information, and is the state of ancillary modes. We suppose that () is an information (ancillary) Majorana fermion. After performing the phase gate , we get the overall output state , and the output state of information Majorana fermions is . In this way, we can realise the single-mode phase operation .

The single-mode phase operation can be performed on a logical Majorana fermion using a sequence of phase operations, i.e. . We only need to operate one ancillary mode, because the number of modes in is odd and each pair of single-mode phase operations is equivalent to a phase gate, i.e. . Such a logical single-mode phase operation commutes with stabiliser operators, therefore it is a fault-tolerant operation. Using logical single-mode phase operations, we can realise the phase gate on logical Majorana fermions, i.e. .

v.2 Initialisation

The initialisation can be realised using a non-destructive measurement and the single-mode phase operation. The initialisation operation reads , where corresponds to the non-destructive measurement, and is a single-mode phase operation depending on the measurement outcome.

Next, we demonstrate how to perform the non-destructive measurement on two logical Majorana fermion modes using the lattice surgery. The lattice surgery is a type of protocols for performing fault-tolerant operations on logical qubits in topological codes Horsman2012 (); Landahl2014 (). In this paper, we propose a set of lattice-surgery protocols to operate logical Majorana fermions. These protocols cannot be derived from lattice-surgery protocols for qubits, because each initialisation and measurement is performed on two vertices rather than one vertex (as in the qubit lattice surgery) on the color code lattice.

v.3 Measurement

There are two types of logical measurements. See Fig. 3(a,b) for the protocols. The lattice of a logical Majorana fermion is a right triangle. If legs of two logical Majorana fermions face each other, we use the type-I measurement; if hypotenuses face each other, we use the type-II measurement.

In order to implement a logical measurement, we need to introduce some ancillary physical Majorana fermion modes. These ancillary modes are initialised at the beginning and measured at the end of the logical measurement. Between initialisation and measurement on ancillary modes, stabiliser operators are repeatedly measured.

The initialisation and measurement pattern are indicated by white bars in Fig. 3(a,b). Each white bar represents a product of two Majorana fermion operators. The pattern is designed to make sure that stabiliser operators of logical Majorana fermions are not damaged in the logical measurement. For example, we consider the stabiliser operator marked by a cross in Fig. 3(a). The stabiliser operator is , in which , , and belong to the logical Majorana fermion , and and are ancillary Majorana fermions. We note that is a stabiliser operator of . Ancillary modes are initialised in the eigenstate with , so at the beginning of the logical measurement we have . At the end of the logical measurement, the eigenvalue of is measured. Then, we have , where the sign is determined by the measurement outcome . We can find that, although the eigenvalue of may be changed by the logical measurement, we can always track its eigenvalue. Therefore, such a stabiliser operator can always be used to detect errors, which is required by the fault tolerance.

Both types of measurements are non-destructive. For the type-I measurement on logical Majorana fermions and , we can find that . Here, is the product of all red stabiliser operators, which is also the product of all Majorana fermion operators; and is the product of all ancillary Majorana fermion operators, which is also the product of all white bars in Fig. 3(a). Given values of and , we can obtain the value of , which is the purpose of the logical measurement. The value of can be obtained by measuring red stabiliser operators. The value of is determined by the initialisation (measurement) on ancillary modes at the beginning (end) of the logical measurement. We suppose that at the beginning, then is the input value of , i.e. the outcome of the logical measurement. Similarly, we also have the output value of at the end, which is . Here, is the product of measurement outcomes of all white bars. The output value may be different from the input value. If they are different, we can change the output value of using a logical phase operation to make sure that the measurement is non-destructive.

For the type-I measurement, the overall operation is described by the superoperator . Here, denotes the initialisation on ancillary modes, and ; denotes stabiliser measurements, and , where is the product of measurement outcomes of all red stabiliser operators; denotes the measurement on ancillary modes, and ; and is a phase operation depending on . Then, we have . Here, we have used that . Therefore, when the value of is , the state of logical Majorana fermions is projected to the eigenstate . In a similar way, we can find that the type-II measurement shown in Fig. 3(b) is also non-destructive, and the outcome of the logical measurement (i.e. the eigenvalue of ) is the product of measurement outcomes of all green stabiliser operators.

The operator is a conserved quantity in the type-I measurement, i.e. , where is the input state. First, the initialisation on ancillary modes does not affect logical Majorana fermions, i.e. . According to the initialisation pattern, (i.e. ), where is the product of all ancillary Majorana fermions in the circle in Fig. 3(a), i.e.  is the product of all white bars on the bottom raw. Then, . Second, stabiliser measurements commute with (which shares even number of modes with each stabiliser operator), so . The product of each pair of white bars on the same blue square is a stabiliser operator, which is a conserved quantity in stabiliser measurements. Because the value of such a blue stabiliser operator is initialised as according to the initialisation pattern, its value is still when white bars are measured. All white bars are paired except white bars on the bottom raw. Therefore, after the measurement on white bars. Third, the effect of the measurement on ancillary modes is , so . Finally, the phase can be cancelled by the phase operation, i.e. . It is similar for the type-II measurement.

v.4 Parity projection

Stabiliser Measurement Parity Projection
Table 1: Measurement-outcome dependent operations in the stabiliser measurement on eight Majorana fermion modes and the logical parity projection. In the stabiliser measurement, the gate is performed depending on measurement outcomes [see Fig. 2(e)]. In the logical parity projection, the operation is performed depending on measurement outcomes . Here, (, and ) is the product of outcomes of white bars on the top raw (bottom raw, left column and right column) in Fig. 3(c). We note that , where is the product of outcomes of all white bars. We remark that gates and are equivalent to each other, and operations and are equivalent to each other.

The logical parity projection shown in Fig. 3(c) is a non-destructive measurement of . We use to denote the overall operation of the logical parity projection. Similar to the type-I measurement, the overall operation includes the initialisation on ancillary modes, stabiliser measurements, the measurement on ancillary modes, and also logical phase operations depending on measurement outcomes. These logical phase operations are listed in Table 1. Analysing the logical parity projection in a similar way to the type-I measurement, we can find that has the following two properties, and , where , is the product of measurement outcomes of all red stabiliser operators, and . Therefore, the logical parity projection projects the state of four logical Majorana fermion modes to the subspace with the eigenvalue .

We can show that is equivalent to the parity projection . Similar to the state of qubits that can be expressed as a polynomial of Pauli operators, the state of fermions can be expressed using fermion operators. The input state can always be written as , which describes three subsystems. The first subsystem is formed by ancillary modes, whose input state is . Majorana fermions for encoding , , and form the second subsystem, and is the projector to the corresponding logical subspace. All other Majorana fermions form the third subsystem, described by operators . The parity projection always projects the input state to a state that can be expressed as , where are functions of . Next, we will show that , i.e. the state of logical Majorana fermions is transformed as the same as in the operation . Here, is a projector describing the final measurement on ancillary modes, is the projector to a new logical subspace, and is a real number. Because the projector to the logical subspace ( or ) and the state of ancillary modes do not carry any logical information, and are equivalent as operations on logical Majorana fermions.

Now, we analyse the effect of on the input state. At the end of , all ancillary Majorana fermions are measured and projected to an eigenstate of all white bars (denoted by the projector ). Similar to the type-I measurement, eigenvalues of stabiliser operators may be changed in the logical operation, i.e. the second subsystem is projected to a new logical subspace at the end of . Therefore , where denotes a state of logical Majorana fermions. The logical subspace of four logical modes is four-dimensional, and projects logical states to a two-dimensional subspace. We can express the two-dimensional subspace as , where are projectors to one-dimensional subspaces. Using properties of , we have . Because are also projectors to one-dimensional subspaces, we have . Using properties of again, we can find that the probability is the same for both signs, and . Then . Using properties of for the third time, we have the output state .

v.5 Magic-state preparation

In order to prepare a magic state in logical Majorana fermions, we propose a scheme that can transfer the magic state of four physical Majorana fermion modes , , and to four logical Majorana fermion modes , , and [see Fig. 3(c,d)]. The magic state is in a two-dimensional subspace, and without loss of generality we suppose that the subspace is . Then, the magic state can always be expressed as , where , and are real numbers. Four physical modes are prepared in the state . Four logical modes are prepared in the state , which can be realised by using a type-I measurement on and followed by a logical parity projection and phase operations if necessary. The lattice for the magic state preparation is the same as the logical parity projection [Fig. 3(c)]. To transfer the magic state, ancillary modes in the central region of the logical parity projection are initialised according to white bars in Fig. 3(d), and other ancillary modes are initialised as the same as in the logical parity projection. Operations after the initialisation are also the same as in the logical parity projection.

The overall operation for transferring the magic state includes the initialisation on ancillary modes (excluding , , and ), stabiliser measurements, the measurement on all ancillary modes, and logical phase operations depending on measurement outcomes. The only difference between and the logical parity projection is the initialisation pattern. To demonstrate that can transfer the magic state, we need to use the following properties, , and . Here, is the product of measurement outcomes of all red stabiliser operators above the central region [see Fig. 3(c,d)], and is the measurement outcome of the white bar . Similar to the logical parity projection, the output state is projected to the subspace , so . We consider the product of all red stabiliser operators above the central region . Measurements on these red stabiliser operators (which are included in ) project the state to the subspace . Because commutes with all white bars in the initialisation pattern, . We note that , where denotes the product of white bars in the initialisation pattern that covered by . Then, due to the initialisation pattern, i.e. . Similar to the logical parity projection, is a conserved quantity in , so . In the logical parity projection, is also a conserved quantity. However, because the initialisation pattern is different, is not a conserved quantity in , and the corresponding conserved quantity becomes , i.e. . The white bar is measured at the end, and we have in the final state, so .

Now, we show that the operation can transfer the magic state from physical Majorana fermions to logical Majorana fermions. We suppose that the input state of all ancillary modes other than , , and is , then the output state reads . Using properties of , we have , where . Similar to the parity projection, we can find that . Therefore, the magic state is transferred to logical Majorana fermions, up to phases that can be corrected using logical phase operations depending on , and .

v.6 Errors in logical operations

Because of the initialisation and measurement pattern, some stabiliser operators cannot be used to detect errors temporarily. We take the type-I measurement as an example [see Fig. 3(a)], and we focus on the region between two dashed lines. At the beginning, after the initialisation on ancillary modes, the value of each green stabiliser operator is determined (which is a product of four white bars), so its outcome in the first round of stabiliser measurements can be compared with its initial value to detect errors. It is similar for blue stabiliser operators. However, for red stabiliser operators, their values cannot be determined by the initialisation pattern, and their outcomes are random in the first round of stabiliser measurements. As a result, red stabiliser operators cannot be used to detect errors at the first round of stabiliser measurements. Similarly, at the end, after the measurement on ancillary modes, we can read values of green and blue stabiliser operators from measurement outcomes of white bars but cannot read values of red stabiliser operators. As a result, we can compare final values of green and blue stabiliser operators with outcomes in the last round of stabiliser measurements to detect errors, but it does not work for red stabiliser operators. Therefore, at the beginning and the end, the examinations of red stabiliser operators are incomplete and cannot provide any information about errors. It is similar for the type-II measurement and the logical parity projection.

Errors may cause incorrect outcomes in stabiliser measurements. We consider a sequence of incorrect measurement outcomes of a stabiliser operator from the beginning to the end. These incorrect outcomes cannot be detected if there are two incomplete stabiliser examinations at the beginning and the end (i.e. we know neither the initial value nor the final value of the stabiliser operator). Such a sequence of incorrect outcomes causes an incorrect outcome of the logical measurement or parity projection, which can be suppressed by repeating stabiliser measurements. The probability of such a logical incorrect outcome decreases with the repetition number of stabiliser measurements between two incomplete stabiliser examinations, the number of incorrect outcomes in the sequence.

The logical phase gate, initialisation, two types of measurements and parity projection are fault-tolerant operations. Logical errors in these operations can be suppressed by enlarging logical Majorana fermions, because the minimum length of nontrivial string operators is  Bombin2006 (). A non-trivial string operator is a string operator connecting boundaries of the code. We remark that incomplete stabiliser examinations are also boundaries.

In the magic-state preparation, the diagonal line of the square lattice [see Fig. 3(d)] separates complete and incomplete stabiliser examinations. According to the initialisation pattern, red (green) stabiliser operators above (below) the diagonal line can be used to detect errors, but red (green) stabiliser operators blew (above) the diagonal line cannot be used to detect errors. This structure makes sure that only errors close to the top-left corner (marked by the bold black square) can cause errors on the output state of logical Majorana fermions Li2015 (), i.e. the rate of errors on the logical magic state is with respect to .

Vi Logical Majorana fermion array and magic-state distillation

Figure 4: (a) Array of logical Majorana fermions. Each right triangle represents a logical Majorana fermion. The red solid line denotes the type-I measurement, and the green dashed line denotes the type-II measurement. The parity projection (PP) can be performed on a group of four logical Majorana fermions (e.g. these four enveloped by the dotted line). The magic state can be prepared on such four logical Majorana fermions. The region in gray are filled by ancillary Majorana fermions. The dimension in the figure does not reflect the actual dimension on the color code lattice. (b) Circuits for the magic state distillation and duplication.

In addition to the set of universal operations, the universal quantum computation also requires the universal connectivity. Any parity-preserving unitary operator on the state of fermions can be achieved using the set of universal operations Bravyi2002 (), under the assumption that a -mode operation can be performed on any set of Majorana fermion modes. Alternatively, we need the ability to transfer the state between any pair of Majorana fermion modes. This requirement leads to the second type of magic states.

We consider the two-dimensional array of logical Majorana fermions shown in Fig. 4(a). Connected by type-I and type-II measurements, logical Majorana fermions form a network. The state of logical Majorana fermions can be transferred among the network with the state transfer operation shown in Fig. 2(b), which can be realised by using logical measurements to initialise and measure logical Majorana fermions. However, the logical measurement can only be performed on a white logical Majorana fermion and a black logical Majorana fermion [see Fig. 4(a)]. In the circuit of the state transfer operation shown in Fig. 2(b), if is a black (white) Majorana fermion, must be a white (black) Majorana fermion, and is also a black (white) Majorana fermion. In other words, the state of a black (white) Majorana fermion can only be transferred to another black (white) Majorana fermion. Therefore, type-I and type-II measurements are not enough for the universal state transfer, and the state transfer is only allowed within a subset of logical Majorana fermions.

In order to transfer the state between two subsets of logical Majorana fermions, we have to introduce the second type of magic states. For four logical Majorana fermions , , and in Fig. 4(a), the magic state is the eigenstate with and . Such a sate corresponds to the qubit magic state (i.e. , and ). Because both and ( and ) are black (white) Majorana fermions, this magic state cannot be prepared using logical measurements. Using and in the magic state to respectively replace and initialised in in Fig. 2(b), we can transfer the state of a white Majorana fermion to a black Majorana fermion . Here is a white Majorana fermion so that the logical measurement can be performed. Similarly, using and in the magic state, we can transfer the state of a black Majorana fermion to a white Majorana fermion. Therefore, this type of magic states completes the universal state transfer required by the universal quantum computation.

Two types of magic states are required, which respectively correspond to qubit states and . Both of them can be prepared using the protocol shown in Fig. 3(d). The prepared magic states are not fault-tolerant, because logical errors in these raw magic states cannot be suppressed by enlarging logical Majorana fermions. In order to obtain magic states with a high fidelity on the fault-tolerant level, we need to distil magic states.

The circuit for the distillation of Y-type magic states is shown in Fig. 4(b). Each round of the distillation needs two copies of the magic state, e.g. prepared on