• Quantum chemistry in the age of quantum computing
  • 5.1 VQE algorithm
  • 5.1.1 Ansatze for state preparation
  • Appendix A: Quantum chemistry basis sets
  • Appendix C: Example of molecular hydrogen
    • Defining the chemistry problem
    • Mapping the problem
    • VQE for quantum chemistry

Quantum chemistry in the age of quantum computing

5.1 VQE algorithm

VQE overview

VQE is an algorithm that uses the variational principle: where wave function preparation and energy estimation are performed on a quantum computer. There are 3 main steps to VQE: 1. State preparation: inital state \(\ket{ \psi(\theta) }\) is prepared by some parameterized ansatz: \(\ket{ \psi(\theta) } = U(\theta) \ket{\psi_0}\). 2. Energy estimation: expectation value of the energy $ $ is estimated using a Hamiltonian averaging procedure, which involes taking measurements of tensor products of Pauli terms. 3. Classical feedback: parameters \(\theta\) are updated using a classical optimizer.

Steps 2 and 3 are repeated until convergence.

5.1.1 Ansatze for state preparation

Random circuits as ansatze fail

1803.11173 helps formalize why choosing the right ansatz is important. They show that for sufficiently random circuits, the variational parameter landscape is plagued by abundance of “barren plateaus,” large regions in which the cost function gradient is nearly 0. [1] What is the cost function gradient? The paper says we can circumvent bad random circuits by using ansatze that have measureable gradients so we can better guide the optimizer.

It’s important to implement ansatze with high-fidelity, low-depth circuits. Maybe I can use Bloch visualizations, but ansatze for the single-qubit states are trivial…

Methods for choosing ansatze

Two main methods in this paper: 1. physically-motivated ansatze (PMA): methods inspired by techniquest that approximate the exact wavefunction 2. hardware heuristic ansatze (HHA): parameterized circuits that use single-qubit rotations and entangling blocks chosen to take advantage of specific hardware

Unitary Coupled Cluster (UCC) ansatz

The best example of a PMA ansatz is the UCC ansatz which “consists of a systematic expansion that approximates the exact many-electon wave function as the excitation rank of the operators in the cluster operator increases.”

We know the UCC ansatz is intractable on a classical computer, but the Trotter-Suzuki decomposition approximation (?) can be implemented efficiently on a quantum computer.

1603.05681 shows UCC can be implemented even in the presence of errors.

The main problem with UCC is that it has many parameters and a long circuit depth.

Coupled-cluster methods

Coupled-cluster methods offer chemically accurate results for many systems and are classically intractable.

Hamiltonian variational ansatz

1507.08969 proposes a Hamiltonian variational ansatz which parameterizes a circuit implementing Hamiltonian simulation for the target Hamiltonian.

Low-depth circuit ansatz

1801.01053 is an ansatz that scales in depth with a circuit to prepare fermionic Gaussian states. LDCA comprises parallel layers of nearest-neighbor matchgates augmented by \(ZZ\) interactions.

Hardware heuristic ansatze structure

HHA’s general structure is as follows: 1. single-qubit rotations 2. fixed entangling operations chosen based on what the hardware can execute best

There isn’t evidence that HHA can efficiently approximate many-electron wave functions.

Appendix A: Quantum chemistry basis sets

Appendix C: Example of molecular hydrogen

Defining the chemistry problem

Problem

We want to determine the ground state energy of \(H\_2\) as a function of distance between nuclei (this is typically known as ground state energy dissociation curve).

Preparing the problem

We apply the Born-Oppenheimer approximation which treats the nuclei as stationary classical particles (which is okay since electron mass:nuclear mass is 1:1000). This gives us the first quantization of the dynamics. First-quantization assumes an infinite Hilbert space, so we must use the second quantization which can be approximated with a finite basis.

First quantization

The Hamiltonian after first quantization is described in atomic units as \[ H_{elec} = - _i - _{i, j} {R_i - r_j } + _{i, j > i} + _{i, j > i} \]

where \(r\_i\) are positions of electrons and \(R\_i\) are positions of nuclei, and \(Z\_i\) and \(M\_i\) denote nuclear charges and masses, respectively.

Second quantization in STO-6G basis

We use the STO-6G basis because it’s the minimal basis.[2] Appendix A has an explanation of mapping to qubits. In this basis, we describe states by occupation of the orbitals.

\[ H = _{pq} h_{pq} a_p^a_q + _{pqrs} h_{pqrs} a_p^a_qa_r a_s \]

where \(a\_p\) are electron annihilation operators that remove an electron from an orbital with label \(p\).

Calculating the \(h\_{pq}\) and \(h\_{pqrs}\) involve some scary integrals that are precomputed on classical computers with electron structure packages.

Mapping the problem

Difference in solving on a classical and quantum computer

Classically, we just solve the second quantization by some more approximations. But on quantum computers, we need an extra “translation” step to encode the second quantization into qubits. We do this by mapping creation and annihilation operators to operators on qubits, where the mapping preserves the algebraic relationships of the operators. [3] Appendix B goes over how to do this mapping.

Bravyi-Kitaev transformation

We use BK transformation, which maps STO-6G second quantization to a four-qubit Hamiltonian: [4] In 1603.05861, they use Jordan-Wigner instead!

\[ ] where the subscript \(j\) indicates what qubit the Pauli operator acts on. The coefficients \(\mu\_k\) are determined by the \(h\_{pq}\) and \(h\_{pqrs}\).

1512.06860 proves that symmetries in this Hamiltonian can be exploited to construct a two-qubit Hamiltonian that contains the ground state. [5] I need to read this paper; I wonder if this is generally reducible or only for \(H\_2\).

The resulting two-qubit Hamiltonian is: \[ H_{elec} = v_0 I + v_1 Z_1 + v_2 Z_2 + v_3 Z_1 Z_2 + v_4 X_1 X_2 + v_5 Y_1 Y_2 \] where \(v\_i\) are linear combinations of \(\mu\_k\).

We will use this to determine the ground state energy as a function of interatomic spacing. [6] Where is the electron spacing in the Hamiltonian? I guess since the second quantization abstracted the bond length away, it’ll all be in the coefficients so I need to find those somehow… The paper above (1512.06860) gives the coefficients in Table 1 on page 13!

VQE for quantum chemistry

Ansatz

The paper chooses the UCC ansatz. They construct it by preparing the Hartree-Fock reference state (\(\ket{01}\)), followed by the UCC operators.

[7] There’s a diagram of UCC ansatz in Fig12 on p149. Implement it.

The paper used L-BFGS-B method to optimize.