Geometric Algebra Jordan–Wigner Transformation for Quantum Simulation

2.1. Clifford Algebra (CA)

A CA is defined using a real vector space, V, with a quadratic form, $\varphi$, over a field [30]. The algebra is produced by the quotient space $T\left(V\right)/I_{\varphi }\left(V\right)$, where T is the tensor product of vector spaces and $I_{\varphi }=v\otimes v-\varphi \left(v\right)$ is the ideal subset closed under multiplication.

For an orthonormal basis, $\{e_1,\cdots ,e_n\}$, spanning the vector space, V, the algebra is generated by the basis verifying

An intuitive definition of a CA can be given using a finite dimensional vector space, V, equipped with the “geometric product” defined as the sum of an inner product and an outer product, which both have a geometrical meaning [21,22], as shown below.

2.2. Mother Algebra and Witt Basis (WB)

The mother algebra is defined from the Euclidean vector space, V, and its dual $V^{*}$ [31]:

The corresponding CA is noted as ${\mathcal{R}}_{n,n}$, called the mother algebra. If $w_i$ is a basis of V and $w_i^{*}$ is a basis of $V^{*}$, then ${\mathcal{R}}_{n,n}$ is the algebra generated by the blades:

with $1\le k,\ell \le n$.

The inner product on ${\mathcal{R}}_{n,n}$ is defined as follows:

Therefore, this set of vectors, $\{w_i,w_i^{*}|1\le i\le n\}$, forms a non-orthonormal basis of the mother algebra. This basis is called the Witt basis (WB).

We can use the following reversion/conjugation properties to transform a blade into its dual when using the WB:

Definition 5.

Reversion/Hermitian conjugation

On a real Clifford space, the Hermitian conjugation is the anti-automorphism of ${\mathcal{C}}_{p,q}$, which maps vectors to their dual:

• $\forall v\in V,v^{†}=v^{*}$ and $v^{*†}=v$

• $\forall x,y\in {\mathcal{C}}_{p,q},{\left(xy\right)}^{†}=y^{†}{x}^{†}$

• $\forall x,y\in {\mathcal{C}}_{p,q},a,b\in \mathbb{R},{(ax+by)}^{†}=a{x}^{†}+b{y}^{†}$

The WB verifies the relations:

An orthonormal basis can be defined from the WB as:

giving $e_i^2=1$ and ${\overline{\varphi }}_i^2=-1$.

Therefore, the mother algebra can be decomposed as a tensor product of two CA, one Euclidean, and the other anti-Euclidean:

where ${\mathcal{R}}_n=\mathcal{C}\ell (e_1,\dots ,e_n)$ and ${\overline{\mathcal{R}}}_n=\mathcal{C}\ell ({\overline{\varphi }}_1,\dots ,{\overline{\varphi }}_n)$. Here, the $e_i$ are Euclidean and the ${\overline{\varphi }}_i$ are anti-Euclidean vectors.

2.3. Witt Basis as a Complex CA

The previous properties can be interpreted as a complex structure of the mother algebra [28,30]. Indeed, by defining the vectors ${\varphi }_j=i{\overline{\varphi }}_j$, where i is the imaginary unit, we obtain the complex CA:

with ${\mathcal{R}}_n^{\prime }=\mathcal{C}\ell ({\varphi }_1,\dots ,{\varphi }_n)$, where now the vectors ${\varphi }_i$ are Euclidean.

In this case, the WB can be written as:

As noted in [31], the anticommutation properties (14) to (19) of the WB are analogous to those of the creation operator, $c^{†}$, and annihilation operator, c, used in quantum physics for fermions, used to write Hamiltonians in the second quantization form.

For this reason, it will be more convenient in the following to use the fermion notation:

2.4. Spinors for Quantum Computing

The WB can be used to build spinor spaces and also to represent qubit states.

Definition 6.

Spinor space

A spinor space is a minimal left ideal of the CA generated by a hermitian idempotent element. It can be written using the mother algebra ${\mathcal{R}}_{n,n}$ of the WB [31]:

$$\begin{array}{c}\hfill S_n={\mathcal{R}}_{n,n}I,\end{array}$$

(28)

where $I\in {\mathcal{R}}_{n,n}$, such that $I^{†}=I$ and is an idempotent $I=I^2$.

The following commuting self-adjoint idempotent operators [32] are then defined:

for $j\in 〚1,n〛$.

The property (14) translates into the closure relation for a single particle:

Then, the spinor ideals can be expressed:

where $\{i_1,\dots ,i_t\}\bigsqcup \{k_1,\dots ,k_s\}=〚1,n〛$ are spinor spaces (the sets are disjoints).

The resolution of the identity is obtained for the entire vector space by using the complete set of idempotent operators:

This means that the entire algebra can be decomposed into a direct sum of $2^n$ spinor spaces.

Quantum computations with n qubits can then be described using these complex CA ${\mathcal{R}}_{n,n}$ by means of the WB, as outlined in [29]; an n-qubit state vector becomes an element of a spinor.

As shown above in (34), the spinor space can be represented by one of the $2^n$ projectors $I_{i_1}\dots I_{i_t}{K}_{k_1}\dots K_{k_s}$. In quantum physics, the natural choice is the empty spinor space, where the projector $I=I_1\dots I_n$ represents the vacuum state $|0\dots 0⟩$.

Using the creation operators, $c_i^{†}$, and the annihilation operators, $c_i$, we have:

In this way, the n-qubit state can be represented in this spinor formulation :

In a more general formulation [32], one could equivalently consider the projector $I_{i_1}\dots I_{i_t}{K}_{k_1}\dots K_{k_s}$ as the original spinor, corresponding now to a qubit state where $i_{\ell }$ qubits are in state |0⟩ and $k_{\ell }$ qubits are in state |1⟩. In this paper, we will adopt the representation of (36).

3.1. Second Quantization

Let us consider a system made of N fermions. The state of the system is described by the state vector

where $x_j$ counts as the number of particles in the state j. Each $x_j$ is either 0 or 1 due to the Pauli exclusion principle.

The annihilation and creation operators, $c_j$ and $c_j^{†}$, give the following relations when applied to $\psi$:

and they correspond to removing or adding a particle in the state j. These operators are not hermitian, so they do not correspond to observable quantities. The number operator

is hermitian and returns the numbers 0 for no-particle and 1 if the particle is present in the state j.

The creation and annihilation operators verify the following properties:

The two first equations give zero because of the Pauli exclusion principle (i.e., $x_j\in \{0,1\}$). The last equation also signifies the closure that

Moreover, the annihilation/creation operators acting on different states always anticommute $\forall i\ne j,$

It is clear that one again finds the same commutation properties as in the case of the WB discussed above in the framework of GA, but here we started exclusively from the quantum theory; this analogy between the two methods justifies our approach.

This representation is called second quantization; it is useful for modeling the Hamiltonians of molecules, where the states represent the possible orbital states for electrons and their spins. This representation can then be used to simulate the molecule by translating this Hamiltonian into quantum gates and applying a quantum algorithm to extract the orbital energies.

Creation and annihilation operators act on fermion states, which is why they anticommute (see (45)), while quantum gates act on qubits, which are represented by wires in a quantum circuit and behave as bosons whose creation and annihilation operators, $a_i^{†}$ and $a_j$, commute. Therefore, fermion creation and annihilation operators and qubit gates do not correspond to the same mathematical structure; this is the reason why a transformation is required to pass from one framework to the other [17].

3.2. Jordan–Wigner Transformation (JWT)

The Jordan–Wigner transformation (JWT) uses the following operators [17,33]:

where the ${\sigma }_j^{+}$ and ${\sigma }_j^{-}$ are spin operators related to the 1-qubit Pauli gates $X_j$ and $Y_j$ by

The inverse transformation is given by:

One verifies that for even products of these operators one finds again the usual 1-qubit projection operators:

The inverse Jordan–Wigner transformation therefore enables one to translate the Hamiltonian of the second quantized form, written with the $c_i^{†}$ and $c_i$ operators into a quantum circuit.

3.3. Permutation Properties of the JWT Operators

When using the WB formalism for the elementary 1-qubit operators $c_j,c_j^{†},c_j{c}_j^{†},c_j^{†}{c}_j$, the permutation of the operators and the qubits states is ruled by the following property:

Proposition 1.

For ${\lambda }_j\in \{c_j,c_j^{†},c_j{c}_j^{†},c_j^{†}{c}_j\}$,

$$\begin{array}{c}\hfill {\lambda }_1|x_1\rangle \dots {\lambda }_n|x_n{\rangle =(-1)}^{{\sum }_{j=1}^n|{\lambda }_j|{\sum }_{k=1}^{j-1}{x}_k}{\lambda }_1\dots {\lambda }_n|x_1\rangle \dots |x_n\rangle ,\end{array}$$

(54)

where $|{\lambda }_j|$ is the number of Witt basis vectors contained in ${\lambda }_j$, (i.e., $|c_j|=1$, $|c_j{c}_i^{†}|=2$, …).

Proof. 

The aim here is to exchange the lambdas and the kets thanks to the anticommutation properties of the WB (44). Then, we only have to prove that

$$\begin{array}{c}\hfill {\lambda }_{j-1}|x_{j-1}\rangle {\lambda }_j={(-1)}^{|{\lambda }_j|x_{j-1}}{\lambda }_{j-1}{\lambda }_j|x_{j-1}\rangle .\end{array}$$

(55)

Depending of the values of ${\lambda }_{j-1}$ and $|x_{j-1}\rangle$, the product ${\lambda }_{j-1}|x_{j-1}\rangle$ takes the values presented in Table 1.

The WB vectors with different indices always anticommute; this means that, in the case of ${\lambda }_j\in \{c_j,c_j^{†}\}$, ${\lambda }_j$ and $|x_{j-1}\rangle$ anticommute if $x_{j-1}=1$ and commute if $x_{j-1}=0$. In the case of ${\lambda }_j\in \{c_j^{†}{c}_j,c_j{c}_j^{†}\}$, ${\lambda }_j$ always commutes with $|x_{j-1}\rangle$. This can be written as (55).

The formula (54) can then be found by exchanging the ${\lambda }_j$ successively with all the previous $|x_k\rangle$.

Let us recall that the operator $n_j$ given in (40) acts on the qubit state as

$$\begin{array}{c}\hfill n_j|x_1\dots x_n\rangle =x_j|x_1\dots x_n\rangle ,\end{array}$$

which gives

$$\begin{array}{c}\hfill {(-1)}^{{\sum }_{j=1}^n|{\lambda }_j|{\sum }_{k=1}^{j-1}{n}_k}|x_1\dots x_n\rangle ={(-1)}^{{\sum }_{j=1}^n|{\lambda }_j|{\sum }_{k=1}^{j-1}{x}_k}|x_1\dots x_n\rangle .\end{array}$$

The above operator can be written using the property ${(-1)}^{{\sum }_{k=1}^{j-1}{n}_k}={\prod }_{j=1}^n(1-2c_j^{†}{c}_j)$.

Then, using (54), we finally get

$$\begin{array}{cc}\hfill {\lambda }_1|x_1\dots {\lambda }_n|x_n\rangle & =\prod _{\begin{array}{c}j=1\\ {\lambda }_j\in \{c_j,c_j^{†}\}\end{array}}^n\prod _{k=1}^{j-1}\left(1-2c_k^{†}{c}_k\right){\lambda }_1\dots {\lambda }_n|x_1\dots x_n\rangle .\hfill \end{array}$$

(56)

□

3.4. Reformulation of Quantum Gates

This result enables one to directly express quantum gates in the GA formalism using the JWT. The problem of sign change due to anticommutativity, which was thoroughly discussed in [29] for the translation of the tensor product of qubits in a GA framework, is here directly encoded inside the JWT.

The JWT applied to the basis matrices leads to:

which can be easily inverted:

Theorem 1.

JWT for 1-qubit gates

The Jordan–Wigner transformation using the Witt basis provides the following expressions for the Pauli 1-qubit gates:

$$\begin{array}{cc}\hfill X_j& =|0⟩{⟨1|}_j+|1⟩{⟨0|}_j=\prod _{k=1}^{j-1}(1-2c_j^{†}{c}_j)(c_j^{†}+c_j)\hfill \end{array}$$

(63)

$$\begin{array}{cc}\hfill Y_j& =-i|0⟩{⟨1|}_j+i|1⟩{⟨0|}_j=\prod _{k=1}^{j-1}(1-2c_j^{†}{c}_j)(i{c}_j^{†}-i{c}_j)\hfill \end{array}$$

(64)

$$\begin{array}{cc}\hfill Z_j& =|0⟩{⟨0|}_j-|1⟩{⟨1|}_j=1-2c_j^{†}{c}_j\hfill \end{array}$$

(65)

This means that, if $\forall j,{\Lambda }_j\in \{X_j,Y_j,Z_j\}$, then we have

$$\begin{array}{c}\hfill {\Lambda }_1|x_1\rangle \dots {\Lambda }_n|x_n\rangle ={\Lambda }_1\dots {\Lambda }_n|x_1\rangle \dots |x_n\rangle .\end{array}$$

(66)

3.5. Two-Qubit Systems

Let us consider the case of a 2-qubit system; this means that we work in the CA ${\mathcal{R}}_{2,2}$. The WB elements reduce to $(w_1^{*},w_1,w_2^{*},w_2)$, here noted as $(c_1,c_1^{†},c_2,c_2^{†})$.

According to the preceding spinor formulation, the qubit states are represented by the following multivectors:

This shows that for 2-qubit states, the Kronecker product is equivalent to the geometric product in ${\mathcal{R}}_{2,2}$. In order to introduce the 1-qubit gates in a 2-qubit circuit, we use the JWT in ${\mathcal{R}}_{2,2}$. For the gates acting on the first qubit, the JWT simply returns:

For the gates acting on the second qubit, however, the JWT adds a correction factor, $c_1{c}_1^{†}-c_1^{†}{c}_1$, to the X and Y gates:

General 2-qubit gates can be written using the sums and products of the above 1-qubit gates. For instance, for the controlled-NOT, controlled-Z, and SWAP gates, we have, respectively:

giving in our formulation in ${\mathcal{R}}_{2,2}$:

3.6. Controlled Gates

In particular, one can derive the expression of controlled gates in ${\mathcal{R}}_{n,n}$. Let $A_k$ be an arbitrary 1-qubit quantum gate acting on the qubit k. The controlled gate $C{\left(A_k\right)}_j$ (where j is the control qubit and k the target) is given by:

4.1. Electronic Hamiltonians for Quantum Simulation

The aim of quantum simulation is to analyze the electronic structure and the associated energies of a molecule. The Born–Oppenheimer approximation (BOA) is commonly used where all nuclei of the molecule are assumed to stay at fixed positions since nuclei have a mass larger than electrons by a factor of ${10}^3$. Therefore, the dynamics of the molecule is described by the electronic Hamiltonian, H, which is the sum of a single electron Hamiltonian, $H^{\left(1\right)}$, and a two-electron Hamiltonian, $H^{\left(2\right)}$. The wave function corresponding to the different energy levels, E (eigenvalues), of the molecule is an eigenfunction $|\psi ⟩$ of the operator H given by the Shrödinger equation:

The ground state, $|{\psi }_g⟩$, corresponds to the lowest energy, $E_g$. To describe electrons in the second quantization, we use a basis of N orbital states, noted as $|{\chi }_j⟩$. Each wave function, ${\chi }_j$, is a single electron wave function and is the tensor product of a spatial component, $|\varphi ⟩$, and a spin component, |↑⟩ or |↓⟩.

In the second quantization formalism, the single-electron and two-electron parts of the BOA Hamiltonians are based only on the relative electron dynamics and are expressed as:

The first coefficient, $h_{ij}$, is the single-electron overlap integral; it consists of the kinetic energy and the potential energy for the one electron interaction from state i to state j. The coefficients $h_{ii}$ correspond to the unperturbed self-energy of the state, $|{\chi }_i⟩$.

The second coefficient $h_{ijkl}$ is the two-electron overlap integral which corresponds to the different interaction energies of a system of 2 electrons initially in states i and j ending up in states k and ℓ. The two-electron interaction comprises : the Coulomb ($i=\ell$, $j=k$) and exchange ($i=k$, $j=\ell$) interactions which represent respectively the interactions between electrons and between electrons and nuclei, the double excitation operator ($i\ne j\ne k\ne \ell$) corresponds to the excitation of both electrons in all four orbital exchanges.

The coefficients $h_{ij}$ and $h_{ijk\ell }$ for a given set of orbitals, $\{|{\chi }_j⟩\}$, can be computed classically by solving the Hartree–Fock equations [17].

4.2. Hamiltonian Operators in the WB

In order to simulate quantum systems using the WB, we express the different contributions to the molecular Hamiltonians; these are shown in Table 2 [20].

In quantum simulations, we are also interested in the exponential of these operators. We will compute them in the following sections. The decomposition of an exponential form is based on the following quantum gates:

and

The index corresponds to the position of the qubit affected by the operator.