A generalized framework for quantum subspace diagonalization

Abstract

We present a framework for computing the solution to Hamiltonian eigenproblems in a subspace defined by bit-strings sampled from a quantum computer. Hamiltonians are represented using an extended alphabet that includes projection and ladder operators, yielding a unified solution method for qubit and fermionic systems. Operators are grouped and sorted so that only non-zero terms are evaluated and a minimal number of subspace lookup operations are performed. Bit-strings are expressed using bit-sets to reduce memory consumption and allow for evaluating operators with no intrinsic limitation on the number of qubits. Subspaces defined over bit-sets are stored in a hash map format that allows for efficient indexing and lookup operations. Our method can be used to directly construct sparse matrix representations or obtain matrix-free solutions. Users are free to utilize these in their eigensolver of choice. We show the benefits of our framework by computing the ground-state solution to examples from condensed matter physics and quantum chemistry with less memory and runtime compared to existing techniques, in some cases by an order of magnitude or more. This work provides a flexible interface for performant quantum-classical eigensolutions for candidate quantum advantage applications.

Figures (4)

• Figure 1: (a) Ratio of qiskit-addon-sqd (SQD) to Fulqrum runtimes for projecting a 1D Heisenberg spin chain Hamiltonian over subspaces defined by post-processed bit-strings sampled from the IBM Marrakesh device for differing numbers of qubits running on Computer A. Inset shows time to solution for the full ground state eigenproblem of the same Hamiltonians and subspaces using SQD (squares-dashed) and Fulqrum (circles-solid). The lowest of five runs is used for timing. (b) Peak memory consumption for full eigensolutions to problems presented in (a).
• Figure 2: (a) Time to solution for computing the ground state eigenproblem to $\mathrm{N}_{2}$ and $\mathrm{CH}_{4}$ dimer using Dice (purple) and Fulqrum for an initial subspace of $\sim 66$ million bit-strings running on Computer B. Times shown are the lowest of five runs. Both Fulqrum (blue) and RAMPS (green) solutions generate CSR matrices using the "fast" Fulqrum CSR workflow, with the latter first trimming the subspace to $1701$ bit-strings before matrix evaluation. The Matrix-free (red) solution uses the full subspace dimensionality. Labels show the elapsed time rounded to the nearest second for brevity. (b) Peak memory consumption for solutions presented in (a). Labels indicate memory consumption rounded to the nearest $100~\rm MiB$.
• Figure 3: Exemplar Fulqrum workflow. Bit-strings sampled from a quantum processing unit (QPU), with optional post-processing such as SQD, are used to construct a Subspace instance that, together with a corresponding qubit (QubitOperator) or fermionic (FermionicOperator) Hamiltonian representation, are used to build a SubspaceHamiltian model. Fermionic systems are cast to equivalent qubit representations via an extended-alphabet JW transformation for a unified solution method. From a SubspaceHamiltonian, users can generate CSR-matrix representations (CSRLinearOperator), or utilize matrix-free solution methods via the abstract LinearOperator class (matvec) in SciPy, passing these on to a wide range of classical eigensolvers arpackcupypolizzi:2009primmeslepc.
• Figure 4: Term sorting methods used in Fulqrum. a) Groups of terms corresponding to matrix elements are sorted into contiguous order by first ordering all terms by their off-diagonal weight, and then sub-sorting terms with matching off-diagonal weight by their off-diagonal structure. Both steps can be done in parallel. The starting and stopping term indices for each group are stored in the group pointers array. b) Sorting by integer value within a group comprised from terms with off-diagonal structure determined by ladder operators.