  Running quantum algorithms often involves implementing complex quantum
circuits with such a large number of multi-qubit gates that the challenge of
tackling practical applications appears daunting. To date, no experiments have
successfully demonstrated a quantum advantage due to the ease with which the
results can be adequately replicated on classical computers through the use of
tensor network algorithms. Additionally, it remains unclear even in theory
where exactly these advantages are rooted within quantum systems because the
logarithmic complexity commonly associated with quantum algorithms is also
present in algorithms based on tensor networks. In this article, we propose a
novel approach called Tensor Quantum Programming, which leverages tensor
networks for hybrid quantum computing. Our key insight is that the primary
challenge of algorithms based on tensor networks lies in their high ranks (bond
dimensions). Quantum computing offers a potential solution to this challenge,
as an ideal quantum computer can represent tensors with arbitrarily high ranks
in contrast to classical counterparts, which indicates the way towards quantum
advantage. While tensor-based vector-encoding and state-readout are known
procedures, the matrix-encoding required for performing matrix-vector
multiplications directly on quantum devices remained unsolved. Here, we
developed an algorithm that encodes Matrix Product Operators into quantum
circuits with a depth that depends linearly on the number of qubits. It
demonstrates effectiveness on up to 50 qubits for several matrices frequently
encountered in differential equations, optimization problems, and quantum
chemistry. We view this work as an initial stride towards the creation of
genuinely practical quantum algorithms.
