More on subspace states

Last year I wrote a post about subspace states, which are a way of efficiently encoding classical data (in the form of d-dimensional subspaces of an n-dimensional vector space) into quantum states using shallow (d log n depth) quantum circuits, by loading one n-dimensional data vector at a time (in log n depth), repeating d times.

After reading the paper, we became interested in whether similar states might be useful for applications beyond quantum machine learning / quantum linear algebra. Luckily for us, the answer is yes!

We have shown that subspace states can also be used to efficiently generate ansatz states for quantum chemistry calculations on quantum processors, in a preprint uploaded to arXiv a few weeks ago.

The idea is to view each data vector as describing the atomic orbitals occupied by an electron, with the total electronic wavefunction built up by adding one electron at a time. For this analogy to work, however, the parity of the electron wavefunction (anti-symmetry under exchange of orbitals) also needs to be encoded; this gives an additional log n overhead compared to encoding classical data. Once parity is taken into account, we can encode any Slater determinant state (i.e. an uncorrelated electronic wavefunction) in a two-qubit gate depth of O(d log^2 n).

Since Slater determinants are easy to classically simulate, a natural question is whether the approach can be adapted to generate more interesting states exhibiting electronic correlation effects (which are what make the electronic structure problem hard to solve). Chee found a way to do this - instead of loading one vector (electron) at a time, pairwise correlations between electrons can be introduced by loading them two at a time. With pairwise correlations, one can generate ansatz states with energies lower than those obtained from the simplest Hartree-Fock approximation, corresponding to a better approximation to the true ground state of the system of interest. The method can also generalize to higher-order correlations.

Some caveats and limitations of our method which didn't make it into the specialist-oriented preprint:

• Gate depth and gate count are two distinct quantities affecting circuit performance; our method has a (poly)-logarithmic two-qubit gate depth, but the number of gates is still linear in the system size n (this is the minimum required to encode all the orbital coefficients). Gate depth matters if the quantum processor has slow two-qubit gates, meaning that only a limited number of gates can be applied in series before the qubits decohere. On the other hand, for fast gates with a lower fidelity, the size of the computation will be limited by the total gate count (too many and an error somewhere is certain). The logarithmic depth arises from using a binary tree to encode the state, meaning that long-range two-qubit gates are required. Altogether, this means that the circuits are best implemented on ion trap systems, which support (slow) long range, high fidelity two-qubit gates.
  
• Advertisements of algorithmic speedups are typically given in terms of better asymptotic scaling. For practical applications, however, an important question is where the actual cross-over will occur, which depends on constant overhead factors present in the implementation. The proposed circuits will be shallower than existing linear depth beam-splitter mesh circuits for system sizes on the order of hundreds of qubits. This means our approach isn't better to run on currently-available cloud quantum processors - we'll need to wait until larger devices come online.
• It's unclear how well our method will perform on near-term devices without quantum error correction. Quantum error mitigation will be needed to get useful results out of any experiment. Since we're not experts in error mitigation we're hoping this functionality will be built-in to future cloud quantum processors.