Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning

We introduce a fermion-to-qubit mapping defined on ternary trees, where any single Majorana operator on an $n$-mode fermionic system is mapped to a multi-qubit Pauli operator acting nontrivially on $\lceil \log_3(2n+1)\rceil$ qubits. The mapping has a simple structure and is optimal in the sense that it is impossible to construct Pauli operators in any fermion-to-qubit mapping acting nontrivially on less than $\log_3(2n)$ qubits on average. We apply it to the problem of learning $k$-fermion reduced density matrix (RDM), a problem relevant in various quantum simulation applications. We show that using the ternary-tree mapping one can determine the elements of all $k$-fermion RDMs, to precision $\epsilon$, by repeating a single quantum circuit for $\lesssim (2n+1)^k \epsilon^{-2}$ times. This result is based on a method we develop here that allows one to determine the elements of all $k$-qubit RDMs, to precision $\epsilon$, by repeating a single quantum circuit for $\lesssim 3^k \epsilon^{-2}$ times, independent of the system size. This improves over existing schemes for determining qubit RDMs.


I. INTRODUCTION
Simulating strongly correlated fermionic systems is hard due to the sign problem [1]. Quantum simulation opens another possibility to solving these hard problems ubiquitous in physics, chemistry, and material sciences (e.g., high temperature superconductivity) [2][3][4][5][6][7]. With the rapid advances in quantum computing devices, such as trapped ions [8][9][10] and superconducting qubits [11,12], we are closer today to realizing a functioning quantum simulator. An essential step in digital quantum simulation of a fermionic systems is mapping it onto a qubit system. Having an efficient, simple, fermion-toqubit mapping is a key ingredient in any quantum simulation protocol, e.g., the variational quantum eigensolver (VQE) [13,14].
The most common fermion-to-qubit mapping is the Jordan-Wigner transformation (JWT) [15], which maps single fermionic operators on an n-mode fermionic system to qubit operators acting nontrivially on O(n) qubits. In the JWT, the qubit state |0 (|1 ) represents an occupied (unoccupied) fermionic mode, i.e., its occupation information. Instead of representing the occupation information of the fermionic modes, one can represent the parity information stored in the fermionic system [16]. In the parity basis, the j-th qubit state |0 (|1 ) stores the information whether the total number of fermions in the first j fermionic modes is even (odd). The Bravyi-Kitaev transformation (BKT) [17] maps single fermionic operators to qubit operators acting nontrivially on O(log 2 n) qubits by balancing the locality of occupation and parity infor-mation. The BKT is important because it avoids operations acting on an extensive number of qubits; however, it is also quite involved, and many authors have discussed its applications in quantum simulation of fermionic systems [16,18]. Recently [19], the BKT was reformulated using the Fenwick trees, a classical data structure that allows for efficient updating elements and calculating prefix sums. It was shown [19] that in this reformulation of the BKT a single fermionic operator is mapped to a Pauli operator acting nontrivially on exactly log 2 n + 1 qubits.
As the possibilities to study fermionic systems on quantum computing devices materialize, it is well timed to explore other, possibly simpler and more efficient fermion-to-qubit transformations, beyond existing ones. Here, we present such a transformation. We construct a simple fermion-to-qubit mapping defined on ternary trees. It maps any single Majorana operator on an nmode fermionic system to a multi-qubit Pauli operator acting nontrivially on log 3 (2n + 1) qubits. For large n, it is approximately log 2 3 1.58 times lower than log 2 n + 1 in the BKT [19]. We prove that the operator weight in the ternary-tree mapping is optimal.
In quantum simulation, it is often desirable to learn the quantum state involving only a fixed numbers of qubits (fermions), i.e., reduced density matrices (RDMs) tomography [20][21][22][23]. Learning the 2-particle RDM of a fermionic system not only allows one to estimate the ground state energy, it also enables one to derive a large number of important properties of the system, such as multipole moments [24] and derivatives of energy [25,26]; moreover, it is an indispensable part in error-mitigating techniques by relaxing the single-particle orbitals [27,28]. Using the ternary-tree mapping, we show that all elements in the k-particle RDM of an n-mode fermionic state stored in a quantum computer can be determined to precision by repeating a single quantum circuit for The technique to obtain the fermionic RDMs is based on a method we develop here that allows one to determine all elements in all k-qubit RDMs of an n-qubit system, to a precision , by repeating a single quantum circuit for < ∼ 3 k / 2 times, independent of the system size n. Our scheme is based on measuring in the Bell basis of a system qubit and an ancilla qubit. It improves the scaling of a recent result [29,30] by a log n factor. Moreover, it is informationally complete, implying that one can retrieve the entire quantum state by simply repeating the same quantum circuit many times.
The paper is organized into two parts. In Sec. II, we introduce the ternary-tree mapping and prove its optimality. In Sec. III, we discuss the Bell-basis measurement scheme for quantum tomography of qubits, and apply it to fermions using the ternary-tree mapping. In App. A, we prove that the Bell-basis measurement scheme for qubits implements a symmetric informationally complete (SIC) POVM. In App. B, we generalize the Bell-basis measurement scheme to qudits and prove its informationally completeness and discuss its relation to SIC POVMs.

II. FERMION-TO-QUBIT MAPPING
In the second quantization formulation to quantum mechanics, the n-mode fermionic operators can be expanded using the Majorana fermion operators for j = 1, 2, . . . , n, where c j and c † j are the j-th annihilation and creation operators. The Majorana operators satisfy the simple anticommutation relation {γ u , γ v } = 2δ uv for u, v = 1, . . . , 2n. In what follows we construct 2n independent multi-qubit Pauli operators that are mutually anticommute, which can be used to represent the Majorana operators.
Our fermion-to-qubit mapping is defined on a ternary tree, where each node (except those in the bottom level of the tree) is associated with a qubit. We start with the case that the tree is complete and come back to the more general cases later on. The total number of qubits in a complete ternary tree of height h is We label the qubit associated with the root node by 0, the rest of the qubits are indexed consecutively, as we are going down the tree, see Fig. 1. The Pauli operators of the η-th qubit are denoted as σ x, y, z η . Each root-toleaf path on the tree can be uniquely specified by the vector p p p = (p 0 , . . . , p h−1 ), where p = 0, 1, 2 determines the next node on the path with depth + 1 based on the current node with depth . We can write the index of a FIG. 1. Ternary-tree mapping. An example of the fermion-to-qubit mapping in a ternary tree of height 2. The label of the qubit associated with a node is written inside it. We introduce a Pauli operator for each root-to-leaf path in the tree. For example, the left-most path corresponds to the Pauli operator A0,0 = σ x 0 σ x 1 .
node of depth on the path p p p by where (3 − 1)/2 is the number of nodes with depth less than . Next, we associate a multi-Pauli operator, A p p p , with each root-to-leaf path p p p, where χ(p) = x, y, z for p = 0, 1, 2, respectively. By construction these operators mutually anticommute, i.e., {A p p p , A q q q } = 0 for p p p = q q q. This is because p p p and q q q have to diverge at some point. Before that point they involve the same Pauli operators on the same qubits, at the point of divergence A p p p and A q q q associate different Pauli operator to the same qubit, whereas after that point they involve Pauli operators on different qubits. There are 3 h = 2n + 1 distinct root-to-leaf paths in the ternary tree, whereas the total number of independent operators A p p p is 2n. This is because the product of A p p p for all paths p p p is proportional to the identity operator. Therefore, we can map 2n Majorana operators to 2n independent Pauli operators that mutually anticommute. The Pauli weight of A p p p equals to the tree height h = log 3 (2n + 1). In the large n limit, it is approximately log 2 3 1.58 times lower than log 2 n + 1 in the BKT [19]. This reduction is achieved by balancing all of the three Pauli operators, whereas only Pauli-x and z are balanced in the BKT.
Consider the case that 2n + 1 is not a power of three, i.e., 3 h < 2n + 1 < 3 h+1 for some nonnegative integer h. We can still construct a ternary-tree mapping by taking the following steps: We now prove that the ternary-tree mapping is optimal.
Theorem 1. The averaged weight of the Pauli operators in any map that transforms an n-mode fermionic system onto a qubit system is at least log 3 (2n).
Proof. We introduce the real vector r = (r 1 , . . . , r 2n ), where r u = γ u = γ † u is the expectation value of the u-th Majorana operator. We also introduce the 2n × 2n Hermitian matrix G whose elements are given by Since G is a Gram matrix it is positive semidefinite. Consider the expectation value where we use γ u γ v = − γ v γ u for u = v and γ u γ u = 1; therefore, we have r r ≤ 1 for any fermionic state. We then consider the system in the state ξ ⊗m , where ξ is the single-qubit pure state which leads to Tr(σ x ξ) = Tr(σ y ξ) = Tr(σ z ξ) = 1/ √ 3. If each γ u is mapped to an m-qubit Pauli operator with weight w u , then we have |r u | = 1/ √ 3 wu under the state ξ ⊗m , and thus where w = 1 2n 2n u=1 w u and we use the fact that y = 3 −x is a convex function in the leftmost inequality. As a consequence, we have When the Pauli weight w u = w is a constant, we have Therefore, the ternary-tree mapping that we have constructed is optimal.

A. Qubits
In this section, we first introduce a method of reconstructing RDMs specifically for qubit systems, and then use this method together with the ternary-tree mapping to reconstruct RDMs for fermionic systems.
Given an n-qubit quantum state ρ, the k-qubit reduced density matrix (k-RDM) may be written as where σ α j is the Pauli operator on the j-th qubit, σ 0 = 1 1 and σ x, y, z are the Pauli-x, -y, and -z operators, respectively. The k-qubit correlation functions in Eq. (11), also referred to as k-RDMs, are defined as Measuring the correlation functions for all α 1 , . . . , α k and all j 1 , . . . , j k amounts to determining all k-RDMs. Assuming all (k−1)-RDMs are known, measuring the n k 3 k different observables, k i=1 σ αi ji (α i = 0), provides us with the required information to reconstruct all k-RDMs. Under the assumption that only single-qubit operations are allowed, Cotler and Wilczek [29] showed that elements in all k-RDMs can be sampled by implementing e O(k) log(n) different quantum circuits. By repeating each circuit 1/ 2 times, the statistical error in estimating these elements scale as .
Changing quantum circuits is typically quite slow on current programmable quantum devices, e.g., those based on FPGAs, while repeating a single circuit may be done much faster. To circumvent the problem of programming various circuits, we propose a scheme that allows one to estimate all k-qubit RDMs at once. Our approach is based on measuring a system qubit and an ancilla qubit in the Bell basis The Bell basis is a common eigenbasis of the commuting operators σ x ⊗ σ x , σ y ⊗ σ y , and σ z ⊗ σ z , and their eigenvalues are listed in Tab. I. Therefore, the expectation values of these three operators can be measured simultaneously. The Bell measurements can be done in parallel for all pairs of qubits using Hardmard and CNOT gates. The quantum circuit to implement our scheme, for one system qubit and one ancilla qubit, is plotted in Fig. 2. If the Pauli-x, -y, and -z operators are sampled at the same rate, a natural choice of the ancilla state is the pure state defined in Eq. (7). It can be obtained by rotating |0 about the x axis by an angle θ = arccos(1/

Preparation of ξ Measurement
where we use Tr(ξσ x ) = Tr(ξσ y ) = Tr(ξσ z ) = 1/ √ 3. Due to the factor 1/ √ 3 k , we must run the experiment 3 k / 2 times to obtain the standard-deviation error . In summery, our scheme measures all k-qubit RMD elements, with error that scales as , by running a single quantum circuit for 3 k / 2 times: 1. To each system qubit (labeled by j) attach an ancillary qubit (labelled by j ) in a known state ξ, so that the total system-ancilla state is ρ ⊗ ξ ⊗n .
The proposed scheme reduces the number of circuit runs by a factor of O(log n) as compared to that of Cotler and Wilczek [29]. While the scheme in [29] only allows one to specify all RDMs up to some order k, our scheme allows for specifying the entire n-qubit state, i.e., all RDMs, for all k ∈ [1, n] [31]. The reason for this inherent difference is rooted in that the measurement we perform on each qubit is informationally complete, that is it allows us to estimate the expectation values of σ x , σ y , and σ z , from one POVM. Since this measurement is done in parallel for all qubits, we can use the data to estimate the entire n-qubit state. In Apps. A, we show it implements a symmetric informationally-complete measurement (SIC POVM) on a qubit, a.k.a., the tetrahedron measurement [32]. Finally, we note that recent work [33] has also suggested to use Bell-measurements in the context of RDMs reconstruction. However, there the use of Bell-measurements was done on pairs of system qubits. In App. B, we generalize the Bell-basis measurement method to D-dimensional system, and show that similar to the qubit case it is informationally-complete and implements SIC POVMs in certain cases.

B. Fermions
Now, we apply the Bell-basis measurement method for RDM reconstruction together with ternary-tree mapping for reconstructing k-fermion RDMs. Generally, it requires O(n 2k ) parameters to determine the k-RDMs of an n-mode fermionic system. To make matters worse, fermions obey anticommutation relations which prevents one from finding large groups of commuting operators that can be measured simultaneously. In second quantization, the elements in a fermionic k-RDM of n modes can be expressed as expectation values involving 2k Majorana operators. For instance, the fermionic 2-RDM may be written as where the γ's are the Majorana fermion operators defined above. Encoded with the ternary-tree mapping, the Pauli weights of elements in fermionic k-RDMs are at most 2k log 3 (2n + 1). With the Bell-basis measurement strategy, the matrix elements are attenuated by a factor bounded by Therefore, all elements in a fermionic k-RDMs can be measured to precision by repeating the same circuit for < ∼ (2n + 1) k / 2 times. Recently, Bonet-Monroig, Babbush, and O'Brien [30] showed that if one is able to implement a linear depth circuit on a linear array prior to measurement, then one can directly measure the fermionic 2-RDM using O(n 2 ) circuits. This result, albeit different from ours, suggests the same scaling of measurement repetitions to sample elements in fermionic RDMs on a quantum computer.

IV. CONCLUSION
The fermionic anticommutation relation has many important implications which include the Pauli exclusion principle. It also underlies intricate constraints on the marginals of the fermionic states, known as the nrepresentability problem [34]. As a direct consequence of the anticommutation relation, we prove a lower bound on the weights of Pauli operators to represent single fermionic operators of an n-mode fermionic system. We constructed a fermion-to-qubit mapping that saturates this bound based on ternary trees, where the numbers of Pauli-x, -y, and -z operators are balanced. We expect the ternary-tree mapping to be a useful tool to studying fermionic systems, such as Hubbard-like systems, on noisy and fault-tolerant quantum computing devices.
Mappings between different types of physical systems are useful, because knowledge about one type of system can be used to understand another type of system, e.g., the properties of certain interacting spin systems can be studied by mapping them to fermions [35]. Another example is the Tomonaga-Luttinger liquid, where certain interacting one-dimensional fermionic systems can be modelled as bosonic systems [36,37]. The ternarytree mapping is both simple and optimal; therefore, it can be a useful tool to study fundamental properties of fermionic systems.
Related to the fermion-to-qubit mapping, we discuss the measurement of fermionic RDMs which is likely to be a bottleneck in quantum simulation of fermions; the noncommuting feature of fermionic operators forbids one to measure them simultaneously. The fermionic anticommutation relation again puts a lower bound on the measurement repetitions required to determine the RDMs. Here, we have shown that all elements in the k-RDM of a fermionic state-stored in a quantum computer with the ternary-tree mapping-can be determined to precision by repeating a single quantum circuit for < ∼ (2n + 1) k / 2 times. In comparison, using the Bravyi-Kitaev transformation leads to < ∼ (n + 1) k log 2 3 / 2 repetitions, which has a worse exponent.
These measurement results for fermions are based on a Bell-basis method we develop here that allows for determining the elements of all k-qubit RDMs, to precision , by repeating a single quantum circuit for < ∼ 3 k −2 times, independent of the system size. This is especially suited for quantum computers with slow circuit updating rates, e.g., those based on FPGAs. Our scheme also saves a log n factor compared to the results by Cotler and Wilczek [29], at the cost of introducing extra ancilla qubits. Another desired property of our scheme is that it is informationally complete, allowing for reconstruction of the entire quantum state as opposed to the RDMs up to some order k. We have also generalized the Bell-basis measurement scheme to qudits using the Heisenberg-Weyl operators.
Future work includes designing fermion-to-qubit mappings on ternary trees with varying depths of root-toleaf paths. This is useful to problems where some of the Majorana operators are more important and need to be sampled more often. They can be mapped to Pauli operators with less weights, corresponding to low-depth paths in the tree. Another interesting line of research is to combine the ternary-tree mapping with quantum error-correcting codes by introducing some redundant qubits [38][39][40].
Proof. The Bell basis is a common eigenbasis of σ x ⊗ σ x , σ y ⊗ σ y , and σ z ⊗ σ z . Therefore, in the limit of infinite number of measurement repetitions, for any k ∈ [1, n], α i = x, y, z and j i ∈ [1, n] (j i = j i for i = i , i, i = 1, . . . , k), using the procedure described in the main text we can calculate the 2k-body correlation function Tr(σ α1 j1 ⊗ σ α1 Since we assumed ξ is known and Tr(σ α ξ) = 0 for α = x, y, z, we can write ρ α1,...,α k j1,...,j k = Tr(σ α1 j1 · · · σ α k j k ρ) . (A4) The SIC POVM on a qubit is a collection of four outcomes (POVM elements) We now show that, following Neumark's theorem it can be implemented by attaching an ancilla qubit in a state ξ = 1 2 (1 1 + 1 √ 3 (σ x + σ y + σ z )) and measuring the two qubits in the Bell-basis. In the Bell-basis measurement, the probability to obtain the outcome that corresponds to Φ + is: Therefore the POVM element on the system qubit which corresponds to the Φ + outcome is Similarly we find, Note that ξ is a pure state and therefore so are σ x ξσ x , σ y ξσ y , and σ z ξσ z . To see that a SIC POVM on a qubit, one can verify explicitly that for α = β = 0, x, y, z Tr(σ α ξσ α σ β ξσ β ) = 1 3 where σ 0 = 1 1. The implementation of the tetrahedron measurement on a qubit using the Bell-basis measurement points at a deep connection between dense coding and SIC POVM of a qubit. More explicitly, the protocol of dense coding is based on the relations These relations allow us to write Eqs. (A7)-(A10), which has the structure of Heisenberg-Weyl (HW) group covariant SIC POVMs. Upon fixing ξ to be a fiducial state of the HW group SIC POVM, as we do above, we recover the tetrahedron measurement.
2. Measure each pair of qudits (j, j ) in the common eigenbasis of X f Z g ⊗ X f Z −g , i.e., the generalized Bell basis.
By choosing a qudit ancilla state ξ such that [41] Tr X fi we obtain The last equation implies that to obtain a fixed standard-deviation error in the measured quantities we must run the experiment (D + 1) k / 2 times, independently of n. In Apps. A, we show that this measurement scheme implements a SIC POVM on a qudit. For example, for a qutrit (three-dimensional system) the pure state ψ = 1 √ 2 (0, 1, −1) is known to be a fiducial state of the HW group covariant SIC POVM [43]. For the proposed protocol to success we only require that Tr(X fi The parameter δ determines the number of total measurement rounds in the protocol. Therefore, for dimensions where the HW group covariant SIC POVMs are unknown or if we are only required to estimate specific correlation functions we can replace the condition (B10) with a less stringent one.

Theorem 3. If
Tr(X f Z −g ξ) = 0 for f, g = 0, . . . , D − 1, then in the limit of infinite number measurement repetitions the above procedure is informationally complete for ρ.
Proof. Since the generalized Bell basis is a common eigenbasis of X f Z g ⊗ X f Z −g , for infinite number measurement repetitions, for any k ∈ [1, n], f, g = 0, . . . , D − 1 and j i ∈ [1, n] (j i = j i for i = i , i, i = 1, . . . , k), using the above procedure we can calculate the 2k-body correlation function Tr X f1 j1 Z g1 j1 ⊗ X f1 Since we assumed ξ is known and Tr X fi j i Z −gi j i ξ = 0 for f i , g i = 0, . . . , D − 1, we can write ρ f1,h1...,f k ,h k j1,...,j k = Tr X f1 j1 Z g1 j1 · · · X f k j k Z g k j k ρ (B12) = Tr X f1 j1 Z g1 j1 ⊗ X f1 This completes the proof.
The SIC POVM on a qudit, a D-dimensional Hilbert space, is a collection of D 2 POVM elements E i = 1 D |ψ i ψ i |, i = 1, . . . , D 2 , such that We now show that, following Neumark's theorem it can be implemented by attaching an ancilla qudit is a state ξ (to be determined later) and measuring the two qudits in a generalized Bell-basis. Consider the following generalization of the Bell-basis to qudits. We define, where X and Z are the HW shift and phase (clock) operators, respectively, where d = 0, . . . , D − 1 and ⊕ is addition modulo d. Following the HW commutation relation, we obtain, where the δ , = 1 if = 0 and δ , = 0 otherwise (and similarly for δ h,h ). therefore, the set {|Φ h : h = 0, . . . , D − 1; = 0, . . . , D − 1} is a set of orthonormal basis (generalized Bell basis) on two qudits.
Following the same calculation done for two-qubit Bell measurement, it is straight forward to check that measuring the generalized Bell basis basis on a two-qudit state ρ ⊗ ξ, defines an effective measurement on the first qudit with d 2 POVM elements