Quantum Operation of Fermionic Systems and Process Tomography Using Majorana Fermion Gates

Quantum operation of fermionic systems and process tomography using Majorana fermion gates

Gang Zhang,1 Mingxia Huo,2 and Ying Li3, ∗ 1College of Physics and Materials Science, Tianjin Normal University, Tianjin 300387, China 2Department of Physics and Beijing Key Laboratory for Magneto-Photoelectrical Composite and Interface Science, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China 3Graduate School of China Academy of Engineering Physics, Beijing 100193, China Quantum tomography is an important tool for the characterisation of quantum operations. In this paper, we present a framework of quantum tomography in fermionic systems. Compared with qubit systems, fermions obey the superselection rule, which sets constraints on states, processes and measurements in a fermionic system. As a result, we can only partly reconstruct an operation that acts on a subset of fermion modes, and the full reconstruction always requires at least one ancillary fermion mode in addition to the subset. We also report a protocol for the full reconstruction based on gates in Majorana fermion quantum computer, including a set of circuits for realising the informationally-complete state preparation and measurement.

I. INTRODUCTION which is similar to the Pauli transfer matrix representation for qubit systems. Using this representation, quan- Majorana fermions are candidates for realising the tum tomography protocols, such as the gate set tomog- topological quantum computation [1]. They are zero- raphy [15–20], can be applied to fermionic systems. Ac- energy modes or quasi-particle excitations in systems cording to SR, the Majorana transfer matrix of a valid such as topological superconducting nanowires [2,3]. quantum process is block diagonal, and two blocks corre- The evidence of Majorana fermions has been observed in spond to transitions between even-parity and odd-parity the experiment [4], and proposals for realising the quan- Majorana fermion operators, respectively. Because valid tum computation operations, e.g. braiding, have been re- quantum states are combinations of only even-parity op- ported [5]. Using Majorana fermions, we can implement erators, we cannot directly measure the odd block in the fermionic quantum computation without the cost of en- process tomography. We show that the odd block can coding fermions in qubits [6,7]. When the noise that be measured by using one ancillary fermion mode (two changes the topological charge, i.e. parity of the parti- Majorana fermion modes). cle number, is sufficiently suppressed, the fault-tolerant For a quantum computation system based on Majorana quantum computation using Majorana fermions is more fermions, we will has a finite set of fermion operations. efficient than using conventional qubits [8]. For instance, We consider an operation set formed by two-mode ini- the encoding cost of the surface-code-based fault tolerant tialisation, measurement, unitary gates, and four-mode quantum computation can be greatly reduced by using entangling operations, which is universal for the quan- Majorana fermions [9]. tum computation using Majorana fermions [6]. We find In this paper, we develop a framework for the quantum that such an operation set is not even sufficient for mea- tomography of fermionic systems. Quantum tomography suring the even block without using ancillary modes. For is an important tool for the verification and characteri- implementing the quantum process tomography in a Ma- sation of quantum operations [10–20]. The tomography jorana fermion quantum computer, we propose tomogra- can measure the full information of a quantum state, pro- phy circuits based on the universal operation set. The cess or measurement. Our result can be used for validat- protocol uses two ancillary Majorana fermion modes, the ing Majorana fermion operations [21] and reconstructing two-mode initialisation, measurement, and exchange gate processes in fermionic quantum computation systems. (the braiding operation). The detailed protocol is illus- arXiv:2102.00620v1 [quant-ph] 1 Feb 2021 Fermionic systems obey the superselection rule (SR): trated in Fig.1. The coherence between states with different particle- number parities is forbidden. Therefore, the quantum tomography of fermionic systems have additional restric- II. FORMALISM OF FERMIONIC QUANTUM tions compared with general quantum systems [22–25]. STATES, PROCESSES AND MEASUREMENTS We first present a set of conditions according to SR that valid quantum states, processes and measurements Fermions are particles that follow the Fermi-Dirac must satisfy. Then, we introduce the Majorana trans- statistics and Pauli exclusion principle. Discrete fermion fer matrix representation by expressing a density matrix modes are described by creation and annihilation oper- as a linear combination of Majorana fermion operators, † ators ai , ai of each mode labelled by i, which obey the † anticommutation relation {ai, aj} = δi,j. The vacuum state |Vi is the state that is empty of particles, i.e. sat- ∗ [email protected] isfies aj|Vi = 0 for all annihilation operators. 2

(a) Tomography circuits (b) Circuit generation A. State

c1 c2 The density matrix of fermion modes is an operator on c3 Hm, which can be expressed in the form c4 Gk U X 0 G M = ρ = ρn,¯ n¯0 |n¯ihn¯ |, (2) -1 Measurement G' n,¯ n¯0 c2m-1 S c2m where ρn,¯ n¯0 are elements of the density matrix. The den- c2m+1 c2m+2 sity matrix must be positive semideﬁnite and normalised, i.e. ρ ≥ 0 and Tr(ρ) = 1. According to SR, [C, ρ] = 0, Initialisation 0 |n¯| |n¯0| Exchange gate i.e. ρn,¯ n¯0 = 0 for alln ¯ andn ¯ that (−1) 6= (−1) [26– (c) Exchange-gate circuits 30]. Theorem 1. The density matrix of fermion modes is valid according to SR if and only if [C, ρ] = 0.

S0 S1 S2 S3 Because C = C† and C2 = 11, the SR condition can be 1 re-expressed as ρ = P(ρ), where P(•) = 2 (• + C • C). FIG. 1. The circuits for the quantum process tomogra- We remark that P is a projection superoperator, i.e. P2 = phy of Majorana fermion modes. (a) To measure the pro- P. For all positive semidefinite and normalised density cess M on 2m Majorana fermion modes c1, c2, . . . , c2m, we matrices ρ, P(ρ) is a valid state. need a pair of ancillary modes c2m+1 and c2m+2. Majorana fermions are initialised pairwisely in the eigenstate of ic2i−1c2i with the eigenvalue +1, where i = 1, 2, . . . , m + 1. Gates 0 B. Process G, G ∈ Gm+1 that act on the 2m + 2 modes are generated according to (b) and (c). After the gate G, the process M is implemented on the first 2m modes. Following the process The physical process of a quantum system is M, the gate U = G0−1 is performed. Then, the final state is characterised by a trace-preserving completely-positive measured pairwisely, i.e. the eigenvalue of each ic2i−1c2i is the map [31]. A completely-positive map can be expressed measurement outcome. (b) The circuit for generating G , P † m+1 as M(ρ) = q FqρFq , where Fq are Kraus operators see Sec.VIA. The inverse of the circuit generates Um+1, see on H [32]. The map is trace-preserving if and only Sec.VIB. G is a 2k-mode gate. To generate a (2k +2)-mode m k if P F †F = 11. A map is unital if M(11) = 11. We gate, the S gate is performed on modes c , c , c and q q q 2k−1 2k 2k+1 note that not all physical processes can be described by c2k+2. To generate a gate G for the state preparation, we take 0 a completely-positive map, specifically when the system S = S0,S1,S2,S3; To generate a gate G for the measurement, we take S = S0,S1,S2. (c) Circuits built on exchanges and environment have initial correlations [33]. gates. Here, S = 11, S = R , S = R and 0 1 c2k,c2k+1 2 c2k,c2k+2 Lemma 1. The output state of a map M is valid ac- S = R2 . 3 c2k,c2k+1 cording to SR for all valid input states if and only if MP = PMP.

† According to SR, for the physical process of fermion Fock states |n¯i = An¯ |Vi form an orthonormal basis of the Hilbert space. Here, modes, the output state of the map must be valid for all valid input states. Lemma1 can be proved by noticing that MP(ρ) = PMP(ρ) for all density matrices ρ. How- n1 n2 nm An¯ = a1 a2 ··· am , (1) ever, this is not the sufficient condition of a valid map on fermion modes, similar to the difference between positive n¯ = (n1, n2, . . . , nm) is a binary vector, ni is the occupa- and completely-positive maps. For a valid fermionic map, tion number of the mode-i and m is the number of modes. if it acts on a subset of fermion modes, the corresponding The Hilbert space of the m modes is Hm = span({|n¯i}). composite map on all the modes should also obey SR. Using Fock states, we can explicitly express states, pro- For a system with m + p modes, the Hilbert space is cesses and measurements of fermion modes. Fermionic Hm ⊗Hp, and we use A and B to denote two subsystems, systems obey the superselection rule (SR): The coherence respectively. The Fock state of the composed system between states with different particle-number parities is 0 0 † † reads |n,¯ n¯ iAB = |n¯iA ⊗ |n¯ iB = AA;¯nAB;¯n0 |ViAB. Note forbidden. Compared with general quantum systems, SR the order of two operators. According to this definition introduces additional restrictions on states, processes and of Fock states for the composite system, we always have measurements of fermion modes. The parity of a Fock aAB;i = aA;i ⊗ 11B if the mode-i is in the first m modes, |n¯| state is (−1) , which is the eigenvalue of the operator where aAB;i is the operator that acts on Hm ⊗ Hp, and Qm † P C = i=1(11−2ai ai). Here, |n¯| = i ni is the Hamming aA;i is the operator that acts on Hm. For a mode-i in weight ofn ¯, which is the particle number. the other p modes, the operator aAB;i cannot be written 3 in the tensor product form with the identity operator on Here, we have used Eq. (5), Eq. (7) and (|ΦiBC is an the subsystem A, to be consistent with the anticommu- eigenstate of CB ⊗ CC) tation relation. Although fermion operators are not local in general, parity operators are local, i.e. CA ⊗ 11B is the hΦ|BCρABCD|ΦiBC parity operator of the first m modes, 11A ⊗ CB is the par- = hΦ|BC(PBC ⊗ IAD)(ρABCD)|ΦiBC. (9) ity operator of the other p modes, and CA ⊗ CB is the parity operator of all modes. The map M denotes a local process on the first m The if part is proved. modes. According to the definition of composite-system A valid state of fermion modes is block diagonal ac- Fock states, the map that acts on the composite system cording to the parity. For a valid map M, although can be expressed in the tensor product form MAB = off-diagonal blocks of the input and output states are MA ⊗ IB, where MA is the map on the operator space zero, the map may have transitions between elements of of Hm, and IB is the identity map on the operator space off-diagonal blocks. An example is the evolution driven of H . We define P (•) = 1 (• + C ⊗ C • C ⊗ C ). by the Hamiltonian H = µa†a for only one mode. The p AB 2 A B A B † † corresponding map is M(•) = e−iµa at • eiµa at. These Theorem 2. A map M on fermion modes is valid ac- off-diagonal-block transitions do not cause any effect on cording to SR if and only if the output state of the mode. However, the effect can be (MA ⊗ IB)PAB = PAB(MA ⊗ IB)PAB, (3) observed in a multi-mode system, e.g. the evolution from the state √1 (|0, 1i + |1, 0i) to √1 (|0, 1i + e−iµt|1, 0i) if for all non-negative integers p. 2 2 the map acts on the first mode. Later, we will show that a valid map according to SR In general, the parity operator C is not a conserved is always local. quantity. For example, the one-mode map M(•) = a • According to Choi’s theorem [31], a map is completely a† + a† • a is valid and flips the parity. The parity is not positive if and only if the corresponding Choi matrix is conserved when the system and environment exchange positive semidefinite. For a fermion map M, the Choi particles. The parity is a conserved quantity in unitary matrix is processes.

Choi(M) = MA ⊗ IB (|ΦihΦ|) , (4) P where |Φi = n¯ |n¯iA ⊗ |n¯iB, and each of A and B has m fermion modes. The map is trace-preserving if and only C. Measurement if TrA [Choi(M)] = 11B, and the map is unital if and only if TrB [Choi(M)] = 11A. The measurement on a quantum system is described by Theorem 3. A map M on fermion modes is valid ac- a set of completely-positive maps {Ek}. Each map can be P † P † cording to SR if and only if [CA ⊗ CB, Choi(M)] = 0. expressed as Ek(•) = q Fk,q • Fk,q, Ek = q Fk,qFk,q P The state |Φi is an eigenstate of the parity operator are POVM operators, and k Ek = 11. Given an input state ρ, the probability of the measurement outcome k is CA ⊗ CB with the eigenvalue +1, i.e. the total particle number is even. Therefore, if M is valid, Tr(Ekρ), and the output state is Ek(ρ)/Tr(Ekρ). For a measurement on fermion modes, maps Ek must Choi(M) = (MA ⊗ IB)PAB (|ΦihΦ|) be valid according to SR. Because Fk,q;A ⊗11B|Φi = 11A ⊗ T = PAB(MA ⊗ IB)PAB (|ΦihΦ|) . (5) Fk,q;B|Φi, where the transpose is in the Fock basis, we have E∗ = Tr [Choi(E )]. For a valid map, [C ⊗ Because Choi(M) = PAB (Choi(M)), we have [CA ⊗ k;B A k A ∗ CB, Choi(M)] = 0. The only if part is proved. CB, Choi(Ek)] = 0, then [CB,Ek;B] = 0. In the Fock Let ρ be a density matrix of m + p modes. Subsystems basis, CB is real. Therefore, [CB,Ek;B] = 0, i.e. [C,Ek] = A, B and C have m modes, and the subsystem D has p 0. modes. According to the teleportation formalism, Theorem 4. A set of POVM operators {Ek} on fermion PAD(ρAD) = hΦ|BC [|ΦihΦ|AB ⊗ PCD(ρCD)] |ΦiBC. (6) modes is valid according to SR if and only if [C,Ek] = 0 Then, for all k.

(MA ⊗ ID)PAD(ρAD) The only if part has been proven. Now we prove = hΦ|BC [Choi(M)AB ⊗ PCD(ρCD)] |ΦiBC. (7) the if part. Given a valid POVM operator√ Ek,√ there Because hΦ|BC •|ΦiBC = hΦ|BCPBC(•)|ΦiBC and (PBC ⊗ exists a completely-positive map Ek(•) = Ek • Ek. IAD)(PAB ⊗ PCD) = (PAD ⊗ IBC)(PAB ⊗ PCD), we have √Because Ek is positive semideﬁnte and block diagonal, Ek is also positive semideﬁnte and block diagonal. We (MA ⊗ ID)PAD(ρAD) √ p have [C, Ek] = 0. Then, |Ψi = Ek;A ⊗ 11B|Φi is an = hΦ| [P (Choi(M) ) ⊗ P (ρ )] |Φi BC AB AB CD CD BC eigenstate of CA ⊗ CB with the eigenvalue +1. Because = PAD(MA ⊗ ID)PAD(ρAD). (8) Choi(Ek) = |ΨihΨ|, Ek is a valid fermion map. 4

III. MAJORANA FERMION OPERATORS In the Fock basis, we can express Majorana fermion operators as [36] Pm Fermion operators can be written in terms of Majorana nj † X j=i+1 1 c2i−1 = ai + ai = (−1) Xi|n¯ihn¯|, (12) fermion operators, ai = 2 (c2i−1 + ic2i). Here, ci are Her- mitian (and unitary) operators that obey the anticom- n¯ Pm † X nj mutation relation {c , c } = 2δ 11. Each fermion mode j=i+1 i j i,j c2i = −i(ai − ai ) = (−1) Yi|n¯ihn¯|. (13) † has to two Majorana fermion operators c2i−1 = ai + ai n¯ † and c2i = −i(ai − ai ), i.e. two Majorana fermion modes. Accordingly, the Jordan-Wigner transformation reads m There are 4 Hermitian operators of m fermion modes m in the product form Y c2i−1 = Xi Zj, (14) j=i+1 C = ib|u¯|/2ccu1 cu2 ··· cu2m , (10) u¯ 1 2 2m m Y c2i = Yi Zj. (15) whereu ¯ = (u1, u2, . . . , u2m) is a binary vector. These operators have properties similar to Pauli operators: j=i+1 We remark that the expression in the Fock basis and • Only the trace of C0¯ = 11 is non-zero, i.e. Tr(Cu¯) = the Jordan-Wigner transformation depend on the deﬁni- m ¯ 2 δu,¯ 0¯, where 0 = (0, 0,..., 0); tion of the Fock state, i.e. the order of fermion operators in A , which must be consistent. When using the Fock 2 n¯ • They are Hermitian and unitary, i.e. Cu¯ = 11; basis and the Jordan-Wigner transformation, we must m take into account all the fermion modes, e.g. including • They are orthogonal, i.e. Tr(C C 0 ) = 2 δ 0 ; u¯ u¯ u,¯ u¯ both m + m modes when we compute the Choi matrix. • They are commutative or anticommutative with each other, i.e. IV. MAJORANA TRANSFER MATRIX

|u¯|·|u¯0|+¯u·u¯0 Cu¯Cu¯0 = (−1) Cu¯0 Cu¯, (11) Similar to the Pauli transfer matrix representation, we can express states, processes and measurements in terms 0 m whereu ¯ · u¯ is the inner product of two binary vec- of Majorana fermion operators Cu¯. These 4 operators tors [34]; are Hermitian and complete (i.e linearly independent). Therefore, we can always express an operator F as a • {±Cu¯, ±iCu¯} is a group, i.e. Cu¯Cu¯0 = ηCu¯+¯u0 , linear combination of Majorana fermion operators, i.e. where η = ±i, ±1, and the + operator of two bi- √ X m nary vectors denotes the element-wise summation F = Fu¯Cu¯/ 2 , (16) modulo 2. u¯ √ m † where Fu¯ = Tr(Cu¯F )/ 2 . If F is Hermitian, coeffi- Because ic2i−1c2i = 2ai ai − 11, the parity operator C = m cients Fu¯ are real; and F¯ = Tr(F ). (−1) C¯, where 1¯ = (1, 1,..., 1). We can find that 0 1 In the Majorana transfer matrix representation, the [C,Cu¯] = 0 for all even-parity operators Cu¯ that |u¯| is m even, and {C,C } = 0 for all odd-parity operators C state is represented by a 4 -dimensional√ column vector u¯ u¯ m that |u¯| is odd. |ρii with real elements |ρ√iiu¯ = Tr(Cu¯ρ)/ 2 . Because ρ m is normalised, |ρii0¯ = 1/ 2 . Similarly, a measurement m operator is represented by a 4 -dimensional√ row vector hhE| with real elements hhE| = Tr(C E)/ 2m. Jordan-Wigner transformation u¯ u¯ According to SR, states and measurement operators obey [C, ρ] = [C,E] = 0. Therefore, |ρiiu¯ = hhE|u¯ = 0 Using the Jordan-Wigner transformation, we can ex- for allu ¯ with odd |u¯|. We define the projections onto press fermion operators using Pauli operators [35], which two 4m/2-dimensional subspaces with even and odd |u¯|, can be used to obtain the explicit expressions of fermion receptively, and they are operators in the Fock basis. We decompose the Hilbert X space of m fermion modes into m subsystems, i.e. Hm = Peven = |u¯iihhu¯|, (17) ⊗m H1 , and the Hilbert space of each subsystem is two- u¯ : |u¯|∈Even dimensional. Accordingly, the Fock state |n¯i = N |n i. X i i P = |u¯iihhu¯|. (18) The four Pauli operators of one subsystem are σI = odd |0ih0| + |1ih1|, σX = |1ih0| + |0ih1|, σY = i|1ih0| − i|0ih1| u¯ : |u¯|∈Odd and σZ = |0ih0| − |1ih1|. We define the Pauli operator Lemma 2. In the Majorana transfer matrix representa- I⊗(i−1) S I⊗(m−i) on m subsystems Si = σ ⊗ σ ⊗ σ , which tion, for valid states ρ and measurement operators E of is the operator of σS = σX , σY , σZ that acts on the i-th fermion modes, the corresponding vectors |ρii and hhE| subsystem. are in the even subspace, i.e. Podd|ρii = hhE|Podd = 0. 5

Matrix representations of a map V. QUANTUM TOMOGRAPHY OF FERMION MODES We can express a completely positive map as X With the Majorana transfer matrix representation and M(•) = χu,¯ u¯0 Cu¯ • Cu¯0 . (19) the conditions according to SR, we can use conven- u,¯ u¯0 tional quantum tomography protocols to implement the The corresponding Choi matrix is tomography on fermion modes. There are two differ- X ences. First, the quantum states, processes and mea- Choi(M) = χu,¯ u¯0 Cu¯|ΦihΦ|Cu¯0 . (20) surements reconstructed in the tomography need to sat- u,¯ u¯0 isfy the SR conditions in addition to other physical conditions. Second, because states and measurements are States Cu¯|Φi are orthogonal, because hΦ|Cu¯Cu¯0 |Φi = even in the subspaces Peven, only the even block M = Tr(Cu¯Cu¯0 ). The particle-number parity of Cu¯|Φi is the m PevenM Peven can be directly measured. parity of |u¯|. Therefore, for a valid map, χu,¯ u¯0 = 0 for allu ¯ andu ¯0 with different parities, i.e. χ is block di- In this section, we first discuss the tomography proto- agonal. Similar to the case of Pauli operators, the map cols for measuring the even block of a map, and then we odd m is completely positive if and only if χ ≥ 0; the map is prove that the odd block M = PoddM Podd can be trace-preserving if and only if Tr(χ) = 1. measured by introducing an ancillary mode (two Majo- rana fermion modes). Theorem 5. A map on fermion modes is valid according to SR if and only if PoddχPeven = PevenχPodd = 0. Now, we can prove that valid maps according to SR A. Quantum tomography of the even block are local. We consider four fermion operators Cu¯1 , Cu¯2 ,

Cu¯3 and Cu¯4 that act on a composite system with m + p The gate set tomography is a self-consistent process modes. In the four operators, Cu¯1 and Cu¯2 act on the ﬁrst tomography protocol [15–20], which does not require the m modes, and Cu¯3 and Cu¯4 act on the other p modes. prior knowledge on the state preparation and measure- A process on the ﬁrst m modes has terms in the form ment. In this section, we take the gate set tomography as

Cu¯1 • Cu¯2 ; and a process on the other p modes has terms an example. Other quantum tomography protocols, e.g. in the form Cu¯3 • Cu¯4 . Note thatu ¯1 · u¯3 =u ¯2 · u¯4 = 0 state tomography and measurement tomography, can be because these operators act on different modes. Then, applied to fermionic systems in a similar way. the composite process has terms in the form Cu¯3 Cu¯1 • To implement the gate set tomography, we need Cu¯2 Cu¯4 = Cu¯1 Cu¯3 • Cu¯4 Cu¯2 . Here, we have used that to prepare a set of linear-independent states |ρiii Cu¯1 and Cu¯2 have the same parity, and Cu¯3 and Cu¯4 and have a set of linearly-independent measurement m have the same parity. Therefore, the process on the first operators hhEk|, where i, k = 1, 2,..., 4 /2. We m modes and the process on the other p modes are always note that 4m/2 is the maximum number of linearly- commutative. independent vectors, limited by dimension of the even subspace. These vectors form two 4m/2-dimensional Theorem 6. Two valid maps M1 and M2 that act on matrices Min = [|ρ1ii |ρ2ii · · · |ρ4m/2ii] and Mout = disjoint modes are always commutative, i.e. [M1, M2] = T T T T 0. [hhE1| hhE2| · · · hhE4m/2| ] . The Gram matrix is g = MoutMin, and each element The Majorana transfer matrix of a map reads of the Gram matrix gk,i = hhEk|ρiii = Tr(Ekρi) can be m −m measured in the experiment, by preparing the state ρi Mu,¯ u¯0 = 2 Tr [Cu¯M(Cu¯0 )] and measuring the probability of the measurement oper- −m X = 2 χv,¯ v¯0 Tr (Cu¯Cv¯Cu¯0 Cv¯0 ) . (21) ator Ek. If Mout is known, we can obtain the prepared v,¯ v¯0 −1 states by computing Min = Moutg. The state tomography can be implemented in this way, where M does not The trace is non-zero if and only if Cu¯Cv¯Cu¯0 Cv¯0 = η11, in where η = ±1, ±i is a phase factor. Therefore, for a need to be a square matrix and can have any number of 0 0 ¯ columns. Similarly, if Min is known, we can obtain mea- non-zero term,u ¯ +v ¯ +u ¯ +v ¯ = 0. Then, we have |u¯ + −1 u¯0| = |v¯ +v ¯0|. Because |v¯ +v ¯0| is even for all non-zero surement operators by computing Mout = gMin . The m 0 measurement tomography can be implemented in this χv,¯ v¯0 , Mu,¯ u¯0 is non-zero only for elements that |u¯ +u ¯ | is even, i.e. the Majorana transfer matrix Mm is also block way, where Mout may not be a square matrix and can diagonal. have any number of rows. We consider a set of maps {Mj}. The matrix of a Lemma 3. Let Mm be the Majorana transfer ma- map Mj that can be directly measured in the experiment trix of a valid fermion map according to SR. Then even m m is Mfj = MoutMj Min. The element of the matrix is PoddM Peven = PevenM Podd = 0. m Mfj;k,i = hhEk|Mj |ρiii = Tr[EkMj(ρi)], which can be The map is trace-preserving if and only if Mm = 0¯,u¯0 measured by preparing the state ρi, implementing the m δ0¯,u¯0 . The map is unital if and only if Mu,¯ 0¯ = δu,¯ 0¯. map Mj and measuring the probability of the measure- 6 ment operator Ek. If Min and Mout are know, we can ob- Next, we will discuss how to measure the odd block. For even tain the even block by computing Mj = MoutMfjMout, simplicity, we assume that Min and Mout are known in which is the process tomography. the following. The gate set tomography can be directly In the gate set tomography, if both Min and Mout generalised to the case of measuring the odd block. are unknown, we can guess an estimate of Mout, which is Mcout, and then compute an estimate of −1 Min, which is Mcin = Mcoutg. The estimate of the even block can be obtained accordingly, which B. The full tomography of a fermion process −1 −1 is Mcj = McoutMfjMcin . These estimates are different from true matrices up to a transformation T = −1 −1 −1 even To measure the odd block of a map M that acts on MoutMcout: Mcin = T Min, Mcj = T Mj T and m modes, we introduce an ancillary mode (two Majorana Mcout = MoutT . The gate set tomography is self- fermion modes). The composite system has m+1 modes, consistent in the sense that McoutMcjN ··· Mcj2 Mcj1 Mcin = the Hilbert space is Hm ⊗ H1, and we use A and B to M Meven ···MevenMevenM for any sequence of denote two subsystems, respectively. Let Mm = Meven⊕ out jN j2 j1 in A maps. Modd be the Majorana transfer matrix of the map on Such a method for implementing the tomography is the operator space of Hm. Then, the Majorana transfer m called the linear inversion method. An alternative way matrix of the map on the composite system reads MAB = even odd is based on the maximum likelihood estimation [15–20]. MAB ⊕ MAB , where

 even  M 0 0 0 Ceven odd even  0 M 0 0  iCoddc2m+1 MAB =  odd  (22)  0 0 M 0  iCoddc2m+2 even 0 0 0 M iCevenc2m+1c2m+2 is the even block of the map on the composite system, and

 odd  M 0 0 0 Codd even odd  0 M 0 0  Cevenc2m+1 MAB =  even  (23)  0 0 M 0  Cevenc2m+2 odd 0 0 0 M iCoddc2m+1c2m+2

is the odd block of the map on the composite system. in the eigenstate of icicj with the eigenvalue +1; ii) The Here, the Majorana fermion operators denote the ba- universal gate set including the exchange gate (braid- 1 π c c sis the matrix, where C = {C : |u¯| ∈ Even} and ing) R = √ (11 + c c ), the gate T = e 8 i j that even u¯ i,j 2 i j i,j Codd = {Cu¯ : |u¯| ∈ Odd} are Cu¯ operators on the first enables non-Clifford qubit gates and the entangling gate π i cicj c cq m modes with the even and odd parities, respectively. Λi,j,k,q = e 4 k ; iii) The measurement on a pair even The even block of the composite system MAB can of Majorana fermion modes to read out the eigenvalue be measured using the quantum tomography, with which of icicj [6]. If the entangling gate is replaced by the we can obtain both Meven and Modd, i.e. the whole Ma- four-mode parity projection, the operation set is still jorana transfer matrix of the map. If we measure the universal. The parity projection is a nondestructive entire even block of the composite system, we will have measurement of the eigenvalue cicjckcq on four Majo- two copies of each Meven and Modd. Later, we will show rana fermion modes, which is described by two maps that one can only measure one copy of each block with- 11+ηcicj ckcq 11+ηcicj ckcq {Mη(•) = 2 • 2 : η = ±1}; Given out using more ancillary modes, because of the limited the input state ρ, when the measurement outcome is the operation set in a Majorana fermion quantum computer. eigenvalue η, the output state is Mη(ρ) up to a normal- isation factor. With the universal operation set, we can implement fault-tolerant universal qubit and fermionic VI. QUANTUM TOMOGRAPHY IN quantum computations [7,9]. In this section, we discuss MAJORANA FERMION QUANTUM how to implement the process tomography in a Majorana COMPUTERS fermion quantum computer using the universal operation set. The universal quantum computation can be implemented on Majorana fermions with the operation set: Given a finite set of fermion modes, we can only mea- i) The preparation of a pair of Majorana fermion modes sure the even block of a map that acts on these modes. If 7 the total number of Majorana fermion modes is 2m, the another basis of the space, which is C+ ∪C−, where C± = m m even block is a 4 /2-dimensional square matrix. To mea- {[11 ± (−1) C]Cu¯ : |u¯| ∈ Even, u2m = 0}. We note that m m m sure the even block, we need to prepare 4 /2 linearly- (−1) CCu¯ = µu¯C1¯−u¯, where µu¯ = (−1) CCu¯C1¯−u¯ = m independent states and have 4 /2 linearly-independent C1¯Cu¯C1¯−u¯ = ±1 is a function ofu ¯. Basis operators in C+ m−1 measurement operators. However, we are not able to pre- and C− are orthogonal and form two 4 -dimensional m m pare 4 /2 linearly-independent states using the limited subspaces. We have CB = BC = ±(−1) B if B ∈ C±. set of operations in a quantum computer. We focus on the operator subspace span(C+). m−1 First, we consider two Majorana fermion modes, c1 and There are at most 4 linearly-independent states 1 c2. The two eigenstates of ic1c2 are ρ1 = 2 (11 + ic1c2) that can be prepared using the limited operation set. We 0 1 suppose these states are {ρ : i = 1, 2,..., 4m−1}, which and ρ1 = (11 − ic1c2), corresponding to eigenvalues +1 i 2 m and −1, respectively. These two states are linearly in- obey Cρi = ρiC = (−1) ρi. Therefore, these states form dependent and sufficient for measuring the even block of a basis of the subspace span(C+), i.e. we can express basis two Majorana fermion modes. However, usually we only operators in terms of states in the form Cu¯ + µu¯C¯ = P 1−u¯ have one initial state in a quantum computer, and let’s i αu,i¯ ρi, where αu,i¯ are real coefficients. assume it is ρ1. We can find that the state ρ1 is invariant Second, we show that 2m+2 Majorana fermion modes under gates S and T . Therefore, given the initial state are sufficient for measuring a map on 2m Majorana ρ1 and the limited set of operations, we cannot measure fermion modes. Let C+ be the set of basis operators the entire even block of two Majorana fermion modes. on 2m + 2 Majorana fermion modes that is accessible for Now, we consider the general case. For 2m Majorana the state preparation. We consider four operators fermion modes, we assume that the initial state of the Qm 1 Gu¯ = Cu¯ + µu¯C1¯−u¯ic2m+1c2m+2, (24) quantum computer is ρm = i=1 2 (11 + ic2i−1c2i), which Hu¯ = C¯ + µu¯Cu¯ic2m+1c2m+2, (25) is the eigenstate of all ic2i−1c2i operators with the same 1−u¯ eigenvalue +1. The state is also an eigenstate of the Iu¯ = Cu¯ic2mc2m+1 + µu¯C1¯−u¯c2mc2m+2, (26) parity operator with the eigenvalue (−1)m, i.e. Cρ = m Ju¯ = C1¯−u¯c2mc2m+1 − µu¯Cu¯ic2mc2m+2, (27) m ρmC = (−1) ρm. We can find that the parity operator is a conserved quantity under the four gates R, T , Λ and the where |u¯| is even and u2m = 0. We have C+ = parity projection. Therefore, we can only prepare states {Gu¯,Hu¯,Iu¯,Ju¯}. Focusing on the first term on RHS of in a subspace of the Hilbert space, i.e. states with the each equation, we can find that Ceven = {Cu¯,C1¯−u¯} and m m−1 parity (−1) . The dimension of the subspace is 2 , so Codd = {Cu¯c2m, −iC1¯−u¯c2m}. In the measurement, if we we can only prepare at most 4m−1 linearly-independent only measure operators of the first two blocks in Eq. (22), density matrices using the limited set of operations. i.e. Ceven and iCoddc2m+1, the second term of each ele- Because of the limited operation set in a Majorana ment in C+ contributes zero to the measurement result. fermion quantum computer, one can only access a sub- In this way, we can reconstruct the first two blocks of space of the Hilbert space. To prepare the total 4m/2 Eq. (22). linearly-independent states, we either need an additional Third, we show that operators Gu¯, Hu¯, Iu¯ and Ju¯ can initial state with a different parity or an additional pair be constructed using the exchange gate. The exchange † † of Majorana fermion modes. With the additional modes, gate has the property Ri,jciRi,j = −cj and Ri,jcjRi,j = we can exchange the particle between two subsystems, ci. We consider the following four states which can flip parities of both subsystems. In this way, 11 + ic2m+1c2m+2 we can effectively prepare an initial state with the differ- ρ = ρ , (28) 4i−3 i 2 ent parity. −1 In summary, given the limited operation set in a quan- ρ4i−2 = R2m,2m+1ρ4i−3R2m,2m+1, (29) −1 tum computer, we are not able to measure the entire ρ4i−1 = R2m,2m+2ρ4i−3R2m,2m+2, (30) even block of a map, although it is allowed according to ρ = R2 ρ R−2 . (31) SR. To measure the entire even block, we need a pair 4i 2m,2m+1 4i−3 2m,2m+1 of ancillary Majorana fermion modes. Note that we also Then, we have need the ancillary pair for measuring the odd block. We X are about to show that with one ancillary pair, we can Gu¯ = αu,i¯ (ρ4i−3 + ρ4i), (32) measure both even and odd blocks. i X Hu¯ = µu¯ αu,i¯ (ρ4i−3 − ρ4i), (33) i A. State preparation protocol X Iu¯ = αu,i¯ (ρ4i−3 + ρ4i − 2ρ4i−1), (34) i First, we identify the operator space which is the span X of states that can be prepared using the limited operation Ju¯ = µu¯ αu,i¯ (ρ4i−3 + ρ4i − 2ρ4i−2). (35) set. We consider 2m Majorana fermion modes. The basis i of the even-parity operator space is Ceven. All valid states Now, we present the state preparation protocol in the according to SR are in this space. Now, we consider inductive way: 8

1. Let G1 be the set of two-mode gates, and G1 = {11} and c2m+2 in the product. We note that all operators in −u2m has only one identity gate; Codd can be expressed as Cu¯i c2m, where Cu¯ ∈ Ceven. We show that these four subsets can be measured by 2. Given the set of 2k-mode gates Gk, we generate the applying exchange gates. Given the measurement of set of (2k + 2)-mode gates [see Fig.1(b) and (c)], Cu¯ ∈ Ceven and the measurement of ic2m+1c2m+2, we can and measure operators Cu¯ and Cu¯ic2m+1c2m+2. If we apply the gate U −1 before measurements, we can eﬀectively −1 −1 Gk+1 = {G, R2k,2k+1G, R2k,2k+2G, measure operators UCu¯U and UCu¯ic2m+1c2m+2U . 2 When u2m = 1, we have R2k,2k+1G : G ∈ Gk}, (36) −1 R2m,2m+1Cu¯R2m,2m+1 = −Cu¯c2mc2m+1, (38) k which has 4 gates; −1 R2m,2m+2Cu¯R2m,2m+2 = −Cu¯c2mc2m+2. (39) 3. Repeat step-2 until Gm+1 is generated; When u2m = 0, we have 4. Prepare the 2m+2 Majorana fermion modes in the R C ic c R−1 initial state 2m,2m+1 u¯ 2m+1 2m+2 2m,2m+1 = Cu¯ic2mc2m+2, (40) m+1 −1 Y 11 + ic2i−1c2i R2m,2m+2Cu¯ic2m+1c2m+2R2m,2m+2 ρ = . (37) 2 = −Cu¯ic2mc2m+1. (41) i=1 Therefore, all operators in the four subsets can be mea- † 5. Generate states ρG = GρG , where G ∈ Gm+1. sured. Similar to the state preparation, we present the mea- In this way, we can prepare 4m linearly independent surement protocol in the inductive way: states of 2m + 2 Majorana fermion modes, which are suﬃcient for implementing the process tomography to 1. Let U1 be the set of two-mode gates, and U1 = {11} measure a map on 2m Majorana fermion modes. has only one identity gate;

2. Given the set of 2k-mode gates Uk, we generate the set of (2k + 2)-mode gates, and B. Measurement protocol −1 −1 Uk+1 = {U, UR , GR : U ∈ Uk}; (42) We need 4m linearly-independent measurement opera- 2k,2k+1 2k,2k+2 tors, which are in the form Ceven and iCoddc2m+1. These operators have the even parity and can be written as a 3. Repeat step-2 until Um+1 is generated; product of two-mode operators icicj. Therefore, we can 4. At the end of the circuit, implement the pairwise realise the measurement straight-forwardly using two- measurements of {ic2i−1c2i : i = 1, 2, . . . , m + 1}. mode measurements. If we only have two-mode measure- With these pairwise measurements, we can effec- ments on specific pairs of modes, we can use the exchange tively measure 2m+1 operators in the form gate to realise the two-mode measurement on any pair of modes. m+1 Now, we show that, using pairwise measurements Y ni Qn¯ = (ic2i−1c2i) . (43) ic2i−1c2i and a subset of inverse gates of G ∈ Gm+1, i=1 we can generate all 4m+1/2 even-parity operators of the 2m + 2 Majorana fermion modes. For only two Majo- 5. Apply the gate U ∈ U before the measurement rana fermion modes, using the measurement of ic c , we m+1 1 2 to effectively measure the operator U †Q U. The can measure two even-parity operators {11, ic c }, where n¯ 1 2 set {U †Q U} includes all even-parity operators of the measurement of the identity operator is trivial. For n¯ the 2m + 2 Majorana fermion modes. 2m Majorana fermion modes, the set of even-parity op- −1 erators is Ceven. These even-parity operators can always We can find that U ∈ Gk for all U ∈ Uk. be measured using pairwise measurements and exchange gates. For 2m + 2 Majorana fermion modes, even-parity op- C. The protocol of tomography erators can be divided into four subsets [see Eq. (22)]: Cu¯ ∈ Ceven, which are operators without c2m+1 and The process tomography of the map M on 2m Majo- 1−u2m c2m+2 in the product; Cu¯i c2mc2m+1, which are op- rana fermion modes can be measured as shown in Fig.1: 1−u2m erators with c2m+1 in the product; Cu¯i c2mc2m+2, which are operators with c2m+2 in the product; and 1. To measure the map on 2m modes, we need two Cu¯ic2m+1c2m+2, which are operators with both c2m+1 ancillary Majorana fermion modes; 9

2. Generate the gate set Gm+1 according to the cir- isfy according to SR. We introduce the Majorana trans- cuit in Fig.1(b) and (c), and circuits are generated fer matrix representation, such that tomography pro- using four gates S = S0,S1,S2,S3; tocols can be applied to fermionic systems. According to SR conditions, we ﬁnd that the full reconstruction 3. Generate the gate set Um+1 formed by inverse of fermion processes always needs at least one ancillary −1 gates U = G , where G ∈ Gm+1 are generated fermion mode (two Majorana fermion modes). In a Ma- by three gates S = S0,S1,S2; jorana fermion quantum computer, the informationally- 4. Prepare each pair of Majorana fermion modes in complete state preparation and measurement can be re- the eigenstate of ic c with the eigenvalue +1; alised using two-mode initialisation, measurement and 2i−1 2i braiding operations, and the protocol is explicitly pre- 5. Choose a gate G ∈ Gm+1 and apply the gate on sented. Our results can be used for the validation of 2m + 2 modes; quantum operations and reconstruction of processes in Majorana fermion systems. 6. Apply the map on the ﬁrst 2m modes;

7. Choose a gate U ∈ Um+1 and apply the gate on 2m + 2 modes;

8. Measure the operator ic2i−1c2i for each pair of Ma- jorana fermion modes. Data generated using these circuits are informationally complete and suﬃcient for computing both the even and ACKNOWLEDGMENTS odd blocks of the map. The state preparation and measurement circuits are simulated numerically using the code available at We thank Wenyu Wu for discussions on the case with https://github.com/Zhanggangtjnu/FMQPT.git. particle number conservation. This work is supported by the National Natural Science Foundation of China (Grant No. 11705127, 11574028, 11874083 and 11875050). GZ VII. CONCLUSIONS acknowledges the support of Program for Innovative Research in University of Tianjin (Grant No. TD13- In this paper, we present the conditions that valid 5077). YL acknowledges the support of NSAF (Grant fermion states, processes and measurements must sat- No. U1930403).

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