Applications of qubization in chemistry

Overview

Phase estimation, as proposed by Kitaev in 1996, identifies the lowest energy eigenstate ( ${\displaystyle |E_{0}\rangle }$ ) and excited states ( ${\displaystyle |E_{i}\rangle }$ ) of a physical Hamiltonian, as detailed by Abrams and Lloyd in 1999.^[6] In quantum computational chemistry, this technique is employed to encode fermionic Hamiltonians into a qubit framework.^[7]

Initialization

The standard quantum phase estimation circuit utilizes three ancilla qubits. In this configuration, when the ancilla qubits are in the state ${\displaystyle |k\rangle }$, a controlled rotation, denoted as ${\displaystyle e^{-\pi \cdot H\cdot i\cdot 2^{k}}}$, is applied to the target state ${\displaystyle |\psi \rangle }$. This operation is a key component of the process. The term 'QFT' refers to the quantum Fourier transform, a fundamental quantum computing operation detailed by . In the final step of the process, the ancilla qubits are measured in the computational basis. This measurement causes the ancilla qubits to collapse to a specific eigenvalue of the Hamiltonian (${\displaystyle H}$), simultaneously collapsing the register qubits into an approximation of the corresponding energy eigenstate. This mechanism is central to the functioning of the quantum phase estimation circuit, allowing for the estimation of energy levels of the system under study.^[8]

The qubit register is initialized in a state, which has a nonzero overlap with the Full Configuration Interaction (FCI) target eigenstate of the system. This state ${\displaystyle |\psi \rangle }$ is expressed as a sum over the energy eigenstates of the Hamiltonian, ${\displaystyle |\psi \rangle =\sum _{i=1}c_{i}|E_{i}\rangle }$ , where ${\displaystyle c_{i}}$represents complex coefficients.^[9]

Application of Hadamard gates

Each ancilla qubit undergoes a Hadamard gate application, placing the ancilla register in a superposed state. Subsequently, controlled gates, as shown above, modify this state.^[9]

Inverse quantum fourier transform

This transform is applied to the ancilla qubits, revealing the phase information that encodes the energy eigenvalues.^[9]

Measurement

The ancilla qubits are measured in the Z basis, collapsing the main register into the corresponding energy eigenstate ${\displaystyle |E_{i}\rangle }$ based on the probability ${\displaystyle |c_{i}|^{2}}$.^[9]

Requirements

The algorithm requires ${\displaystyle \omega }$ ancilla qubits, with their number determined by the desired precision and success probability of the energy estimate. Obtaining a binary energy estimate precise to n bits with a success probability ${\displaystyle p}$ necessitates.^[9]${\displaystyle \omega =n+\lceil \log _{2}\left(2+{\frac {1}{2p}}\right)\rceil }$ ancilla qubits. This phase estimation has been validated experimentally across various quantum architectures.^[9]

Time evolution and error analysis

The total coherent time evolution ${\displaystyle T}$ required for the algorithm is approximately ${\displaystyle T=2^{(\omega +1)}\pi }$.^[10] The total evolution time is related to the binary precision ${\displaystyle \varepsilon _{\text{PE}}={\frac {1}{2^{n}}}}$, with an expected repeat of the procedure for accurate ground state estimation. Errors in the algorithm include errors in energy eigenvalue estimation (${\displaystyle \varepsilon _{PE}}$), unitary evolutions (${\displaystyle \varepsilon _{U}}$), and circuit synthesis errors (${\displaystyle \varepsilon _{CS}}$), which can be quantified using techniques like the Solovay-Kitaev theorem.^[11]

The phase estimation algorithm can be enhanced or altered in several ways, such as using a single ancilla qubit  for sequential measurements, increasing efficiency, parallelization, or enhancing noise resilience in analytical chemistry. The algorithm can also be scaled using classically obtained knowledge about energy gaps between states.^[12]

Limitations

Effective state preparation is needed, as a randomly chosen state would exponentially decrease the probability of collapsing to the desired ground state. Various methods for state preparation have been proposed, including classical approaches and quantum techniques like adiabatic state preparation.^[13]

Overview

The variational quantum eigensolver is an innovative algorithm in quantum computing, crucial for near-term quantum hardware.^[14] Initially proposed by Peruzzo et al. in 2014 and further developed by McClean et al. in 2016, VQE is integral in finding the lowest eigenvalue of Hamiltonians, particularly those in chemical systems.^[15] It employs the variational method (quantum mechanics), which guarantees that the expectation value of the Hamiltonian for any parameterized trial wave function is at least the lowest energy eigenvalue of that Hamiltonian.^[16] This principle is fundamental in VQE's strategy to optimize parameters and find the ground state energy.  VQE is a hybrid algorithm that utilizes both quantum and classical computers. The quantum computer prepares and measures the quantum state, while the classical computer processes these measurements and updates the system. This synergy allows VQE to overcome some limitations of purely quantum methods.^[17]

Applications of VQEs in chemistry

Overview

In quantum chemistry, electrons are modeled as fermions with antisymmetric wave functions. The Jordan-Wigner encoding maps these fermionic orbitals to qubits, preserving their antisymmetric nature. Mathematically, this is achieved by associating each fermionic creation ${\displaystyle (a_{i}^{\dagger })}$ and annihilation ${\displaystyle (a_{i})}$ operator with corresponding qubit operators through the Jordan-Wigner transformation:

${\displaystyle a_{i}^{\dagger }\rightarrow {\frac {1}{2}}\left(\prod _{k=1}^{i-1}Z_{k}\right)(X_{i}-iY_{i})}$

Where ${\displaystyle X_{i}}$ , ${\displaystyle Y_{i}}$ , and ${\displaystyle Z_{i}}$ are Pauli matrices acting on the ${\displaystyle i^{\text{th}}}$ qubit.^[23]

Applications of Jordan-Wigner encoding in chemistry

Electron hopping

Electron hopping between orbitals, central to chemical bonding and reactions, is represented by terms like ${\displaystyle a^{\dagger _{i}}a_{j}+a^{\dagger _{j}}a_{i}}$. Under Jordan-Wigner encoding, these transform as follows:^[23]

$${\displaystyle a_{i}^{\dagger }a_{j}+a_{j}^{\dagger }a_{i}\rightarrow {\frac {1}{2}}(X_{i}X_{j}+Y_{i}Y_{j})Z_{i+1}\cdots Z_{j-1}}$$

This transformation captures the quantum mechanical behavior of electron movement and interaction within molecules.^[24]

Limitations of Jordan–Wigner encoding

The Jordan-Wigner transformation encodes fermionic operators into qubit operators, but it introduces non-local string operators that can make simulations inefficient. The FSWAP gate is used to mitigate this inefficiency by rearranging the ordering of fermions (or their qubit representations), thus simplifying the implementation of fermionic operations.^[26]