Gate construction of block-encoding for Hamiltonians needed for simulating partial differential equations
One of the most promising applications of quantum computers is solving partial differential equations (PDEs). By using the Schrödingerization technique—which converts nonconservative PDEs into Schrödinger equations—the problem can be reduced to Hamiltonian simulations. The particular class of Hamilt...
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| Format: | Article |
| Language: | English |
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American Physical Society
2025-07-01
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| Series: | Physical Review Research |
| Online Access: | http://doi.org/10.1103/xlpd-fb1g |
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| author | Nikita Guseynov Xiajie Huang Nana Liu |
| author_facet | Nikita Guseynov Xiajie Huang Nana Liu |
| author_sort | Nikita Guseynov |
| collection | DOAJ |
| description | One of the most promising applications of quantum computers is solving partial differential equations (PDEs). By using the Schrödingerization technique—which converts nonconservative PDEs into Schrödinger equations—the problem can be reduced to Hamiltonian simulations. The particular class of Hamiltonians we consider is shown to be sufficient for simulating almost any linear PDE. In particular, these Hamiltonians consist of discretizations of polynomial products and sums of position and momentum operators. This paper addresses an important gap by efficiently loading these Hamiltonians into the quantum computer through block-encoding. The construction is explicit and efficient in terms of one- and two-qubit operations, forming a fundamental building block for constructing the unitary evolution operator for that class of Hamiltonians. The proposed algorithm demonstrates a squared logarithmic scaling with respect to the spatial partitioning size, offering a polynomial speedup over classical finite-difference methods in the context of spatial partitioning for solving PDEs. Furthermore, the algorithm is extended to the multidimensional case, achieving an exponential acceleration with respect to the number of dimensions, alleviating the curse of dimensionality problem. This work provides an essential foundation for developing explicit and efficient quantum circuits for PDEs, Hamiltonian simulations, and ground state and thermal state preparation. |
| format | Article |
| id | doaj-art-d7eaf535375d4487bc44e29c81fba2f2 |
| institution | Kabale University |
| issn | 2643-1564 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | American Physical Society |
| record_format | Article |
| series | Physical Review Research |
| spelling | doaj-art-d7eaf535375d4487bc44e29c81fba2f22025-08-20T03:31:33ZengAmerican Physical SocietyPhysical Review Research2643-15642025-07-017303310010.1103/xlpd-fb1gGate construction of block-encoding for Hamiltonians needed for simulating partial differential equationsNikita GuseynovXiajie HuangNana LiuOne of the most promising applications of quantum computers is solving partial differential equations (PDEs). By using the Schrödingerization technique—which converts nonconservative PDEs into Schrödinger equations—the problem can be reduced to Hamiltonian simulations. The particular class of Hamiltonians we consider is shown to be sufficient for simulating almost any linear PDE. In particular, these Hamiltonians consist of discretizations of polynomial products and sums of position and momentum operators. This paper addresses an important gap by efficiently loading these Hamiltonians into the quantum computer through block-encoding. The construction is explicit and efficient in terms of one- and two-qubit operations, forming a fundamental building block for constructing the unitary evolution operator for that class of Hamiltonians. The proposed algorithm demonstrates a squared logarithmic scaling with respect to the spatial partitioning size, offering a polynomial speedup over classical finite-difference methods in the context of spatial partitioning for solving PDEs. Furthermore, the algorithm is extended to the multidimensional case, achieving an exponential acceleration with respect to the number of dimensions, alleviating the curse of dimensionality problem. This work provides an essential foundation for developing explicit and efficient quantum circuits for PDEs, Hamiltonian simulations, and ground state and thermal state preparation.http://doi.org/10.1103/xlpd-fb1g |
| spellingShingle | Nikita Guseynov Xiajie Huang Nana Liu Gate construction of block-encoding for Hamiltonians needed for simulating partial differential equations Physical Review Research |
| title | Gate construction of block-encoding for Hamiltonians needed for simulating partial differential equations |
| title_full | Gate construction of block-encoding for Hamiltonians needed for simulating partial differential equations |
| title_fullStr | Gate construction of block-encoding for Hamiltonians needed for simulating partial differential equations |
| title_full_unstemmed | Gate construction of block-encoding for Hamiltonians needed for simulating partial differential equations |
| title_short | Gate construction of block-encoding for Hamiltonians needed for simulating partial differential equations |
| title_sort | gate construction of block encoding for hamiltonians needed for simulating partial differential equations |
| url | http://doi.org/10.1103/xlpd-fb1g |
| work_keys_str_mv | AT nikitaguseynov gateconstructionofblockencodingforhamiltoniansneededforsimulatingpartialdifferentialequations AT xiajiehuang gateconstructionofblockencodingforhamiltoniansneededforsimulatingpartialdifferentialequations AT nanaliu gateconstructionofblockencodingforhamiltoniansneededforsimulatingpartialdifferentialequations |