Preconditioned Block Encodings for Quantum Linear Systems

TL;DR

It is seen that encoding of the classical product can significantly improve the effective condition number using the sparse approximate inverse preconditioner with infill, and a new matrix filtering technique is introduced that reduces the circuit depth without adversely affecting the matrix solution.

Abstract

Quantum linear system solvers like the Quantum Singular Value Transformation (QSVT) require a block encoding of the system matrix $A$ within a unitary operator $U_A$. Unfortunately, block encoding often results in significant subnormalisation and increase in the matrix's effective condition number $κ$, affecting the efficiency of solvers. Matrix preconditioning is a well-established classical technique to reduce $κ$ by multiplying $A$ by a preconditioner $P$. Here, we study quantum preconditioning for block encodings. We consider four preconditioners and two encoding approaches: (a) separately encoding $A$ and its preconditioner $P$, followed by quantum multiplication, and (b) classically multiplying $A$ and $P$ before encoding the product in $U_{PA}$. Their impact on subnormalisation factors and condition number $κ$ are analysed using practical matrices from Computational Fluid Dynamics (CFD). Our results show that (a) quantum multiplication introduces excessive subnormalisation factors, negating improvements in $κ$. We introduce preamplified quantum multiplication to reduce subnormalisation, which is of independent interest. Conversely, we see that (b) encoding of the classical product can significantly improve the effective condition number using the Sparse Approximate Inverse preconditioner with infill. Further, we introduce a new matrix filtering technique that reduces the circuit depth without adversely affecting the matrix solution. We apply these methods to reduce the number of QSVT phase factors by a factor of 25 for an example CFD matrix of size 1024x1024.

Figures (17)

• Figure 1: Circuit for encoding a 16x16 pentadiagonal Toeplitz matrix with diagonals at offsets: $-4$, $-1$, 0, 1, 4. Negative offsets are super-diagonals, positive offsets are sub-diagonals and 0 is the main diagonal. Strictly, the above circuit has a controlled addition of 0, but this is omitted as it has no effect.
• Figure 2: Circuit for encoding a 16x16 pentadiagonal matrix using an LCU to add separately encoded diagonals. Each diagonal encoding operator $D_i$ includes the addition or subtraction of $i$ to give the correct offset from the main diagonal.
• Figure 3: Circuit segment for encoding a banded 8x8 tridiagonal matrix with one super-diagonal offset by 1 from the main diagonal and one sub-diagonal offset by 4 from the main diagonal. Note only 2 prep qubits are needed to load the 3 diagonals.
• Figure 4: Circuit for encoding $C^{-1} A$ for an 8x8 pentadiagonal matrix.
• Figure 5: Classical preconditioning - condition numbers for $PA$ as used by a classical algorithm. Computed using matrix-matrix multiplication. Numbers in brackets indicate levels of infill. All results are pre-scaled as described in \ref{['subsec-encode-orig']} except the 'No Scaling' line in (a).