Resource-Efficient Block Encoding Enables Quantum Algorithms
Block-encoding
Block-encoding operators are vital in quantum signal processing, and Fraunhofer IAO and Universität Stuttgart researchers found a resource-efficient way to construct them. This allows for larger, more sophisticated quantum algorithms and reduces the processing complexity of building them. The unique method assembles these fundamental quantum elements with almost optimal resource requirements for a wide range of input matrices, yielding a parameter count close to the number of free parameters. This crucial innovation enables optimisation for quantum systems with up to eight qubits.
Block-encoding is crucial to quantum signal processing (QSP) methods, which dominate quantum computer research. It allows quantum computers to mimic Hamiltonians in physics and quantum chemistry by representing non-unitary input matrices as sub-blocks of larger unitary operators.
Oracle query models and Linear Combinations of Unitarizes (LCU) can produce quantum advantages, but they often have drawbacks for near-term quantum computers, such as high ancilla overheads, a high number of multi-controlled gates, and unfavourable scaling with dense and unstructured matrices. Block-encoding operator gate complexity usually determines the quantum computing cost of the full quantum procedure.
To tackle these challenges, the group created Variational Block-Encoding (VBE). By encoding matrices with hardware-efficient parameterised quantum circuits (PQCs), VBE promises to improve quantum calculations on low-resource hardware.
Given that conventional methods require more than one ancilla qubit, this study's primary finding is that VBE can encode precisely with one. This matches theoretical bottom circuit parameter limits associated to the target matrix's degrees of freedom.
This study involves adapting the circuit architecture to the input matrix's intrinsic features, such as real or Hermitian values or symmetries. By directly integrating these symmetries into the circuit architecture, researchers can reduce encoding parameters and enhance efficiency. This parameter minimisation is crucial because it closely matches the quantum resources qubits and compute operations. Limiting circuits to real-valued or Hermitian targets reduces resource expenditures.
To build realistic quantum computers with fewer errors, the research explored the expressibility and complexity of quantum circuits, focussing on powerful circuits that can represent a matrix while being simple. Lie algebra and the Derivative Lie Algebra are used to analyse these circuits' generators to determine their capabilities and range of activities.
The generator basis set size determines the circuit's impressibility. The study found a correlation between the algebraic structure of circuit generators and the complexity of symmetry-restricted circuits, enabling more effective encoding schemes.
Numerical studies reveal that VBE efficiently encodes dense input matrices with the quantum circuit's free parameters closely matching the target matrix's lower bound of independent parameters. The smooth optimisation landscapes in VBE allowed classical optimisation methods like the BFGS optimiser to converge. The technique is very durable in the overparameterized setting, where any point can reach a global minimum.
VBE has significantly lower resource overhead than current block-encoding methods. In Heisenberg Hamiltonians, VBE reduces 2-qubit gate counts, a critical quantum resource metric, by more than an order of magnitude compared to LCU for systems up to five sites.
Systems with eight permutation-invariant circuit locations benefit from this. For larger systems, 2-qubit gate counts for LCU may seem smaller since the number of LCU terms increases polynomially, while VBE circuit sizes for most ansatzes save the permutation invariant one increase exponentially.
Despite these advances, VBE's fundamental limitation is the large amount of classical computation needed to tune variational parameters, which limits its application to systems with up to eight qubits. Optimisation landscapes and matrix calculations are exponentially more complex and expensive as systems grow.
The researchers suggest using VBE with LCU (linear combination of unitarizes) in the near future. VBE could encode smaller matrix blocks, and linear combinations could build the whole matrix, reducing resource needs.
Future research will examine multivariate quantum signal processing and how other system-specific factors can reduce circuit resource requirements. These investigations may yield circuit parameter determination methods without optimisation. The strategy also improves quantum machine learning and variational quantum eigensolvers.
