Semiclassical Fourier Transform for Quantum Computation

Recent developments in quantum computation offer a way forward for determining efficient solutions of many instances of large eigenvalue problems that are classically intractable.

“Recent developments in the field of quantum computation offer a way forward for determining efficient solutions of many instances of large eigenvalue problems that are classically intractable6789101112”

The QFT can be implemented semi-classically and interleaved with controlled modular multiplications.

“The other part of the algorithm, the quantum Fourier transform (QFT), can be implemented semi-classically [40] and interleaved with the controlled modular multiplications.”

The construction uses 3n + O(lg n) logical qubits, distributed among accumulation, workspace, lookup output, and exponent registers.

“In terms of space, we are using 3 n + O (lg n ) logical qubits. The qubits are distributed as follows: n + O (lg n ) for the accumulation register storing the product, n + O (lg n ) for a workspace register during the multiplication, n + O (lg n ) for the lookup output register, and O (lg n ) qubits to hold the part of the exponent needed for the current stage of the semi-classical Fourier trans”

Babbush et al. showed that the electronic structure Hamiltonian requires only O(N^2) terms in a plane wave basis while O(N^4) terms are necessary for Gaussians.

“Babbush et al. 301 showed that the electronic structure Hamiltonian requires only O ( N 2 ) terms in a plane wave basis while O ( N 4 ) terms are necessary for Gaussians.”

This allows a representation of the chemistry Hamiltonian with a number of terms scaling quadratically in the number of spin-orbitals, a significant reduction on the number of terms in the Hamiltonian when using molecular orbitals.

“This allows a representation of the chemistry Hamiltonian with a number of terms scaling quadratically in the number of spin-orbitals, a significant reduction on the number of terms in the Hamiltonian when using molecular orbitals.”

Babbush et al. proposed using a dual form of the plane wave basis that diagonalizes the potential operator, leading to a representation of the electronic structure Hamiltonian with only O(N^2) terms.

“In particular, Babbush et al. have proposed to use a dual form of the plane wave basis 301 which diagonalizes the potential operator, leading to the following representation of the electronic structure Hamiltonian that has only O ( N 2 ) terms:”

When combined with the simplified representations of the Hamiltonian, the Hamiltonian variational method can be used to construct variational circuits with depth equal to the number of spin-orbitals for a linear qubit architecture, using the fermionic swap network.

“When combined with the simplified representations of the Hamiltonian described in the previous section 301 , the Hamiltonian variational method can be used to construct variational circuits with depth equal to the number of spin-orbitals for a linear qubit architecture, using the fermionic swap network 302 .”

Kivlichan et al. demonstrated that a single Trotter step of the Hamiltonian in Equation 44 can be implemented with a circuit of depth N using a Fermionic Swap Network, which has a constant improvement with respect to the proposal in Babbush et al. and is conjectured to be optimal.

“Finally, Kivlichan et al. 302 demonstrated that a single Trotter step of the Hamiltonian in Equation 44 can be implemented with a circuit of depth N using a Fermionic Swap Network , which has a constant improvement with respect to the proposal in Babbush et al. 301 and is conjectured to be optimal.”

In Babbush et al., the idea of using the plane wave dual basis with the Hamiltonian variational strategy is developed to propose a VQE simulation for Jellium that requires depth O(N·S) and optimizes only O(N) parameters by exploiting translational invariance.

“In Babbush et al. 301 , this idea is developed to propose a VQE simulation for Jellium, that requires depth O ( N · S ) and optimizes only over O ( N ) parameters by exploiting the translational invariance of this system.”

Since in finite molecules and bulk materials η ∈ O(N), the number of measurements does not scale worse than O(N^4/ε^2) for this representation of the Hamiltonian.

“Since in finite molecules and bulk materials, η ∈ O ( N ), this shows that the number of measurements does not scale worse than O ( N 4 /ϵ 2 ) for this representation of the Hamiltonian.”

For Hamiltonians in the plane wave dual basis, the number of measurements required to estimate the expectation values can be bounded, and further savings are possible because the interaction (V) and potential (U) terms are diagonal and commute, allowing separate unbiased estimation for the mean of their sum.

“Alternatively, for Hamiltonians in the plane wave dual basis (Equation 44), the number of measurements required to estimate the expectation values can be bounded as 301”

The dual plane-wave basis set has been proposed as an alternative for electronic structure calculations on quantum computers.

“Recently, the dual plane-wave basis set (dual PW), corresponding to the Fourier transform of PW, has been proposed as an alternative for electronic structure calculations on quantum computers 301”

The dual PW basis set provides a more compact representation of the molecular Hamiltonian in second quantization.

“The dual PW basis set has the advantage of providing a more compact representation of the molecular Hamiltonian in second quantization.”

Cold atoms representing qubits can be arranged on a two- or three-dimensional lattice and state-dependent interaction phases may be acquired via cold collisions between neighboring atoms or via tunneling.

“Cold atoms representing the qubits can be arranged on a two-or three dimensional lattice and state-dependent interaction phases may be acquired via cold collisions between neighboring atoms [12] or via tunneling [13].”