#hamiltonianembedding

Hamiltonian Embedding: A Quantum Method for Simulating High-Dimensional Dynamics on Near-Term Hardware

A team of researchers from the University of Maryland and the University of California, Berkeley created Hamiltonian embedding to bridge the gap between near-term quantum hardware limits and complicated computational difficulties. This method makes it easier to simulate high-dimensional dynamics governed by partial differential equations (PDEs) that classical computers cannot solve.

Scientists and engineers in fluid dynamics, aircraft design, and heat and sound propagation need PDE simulation. Due to the exponential computer difficulty of solving high-dimensional differential equations, classical skills are severely limited. Many PDEs have been solved using quantum methods, but most require complicated input models like block-encoded matrices and Quantum Random Access Memory (QRAM), which require massive, fault-tolerant quantum computers that are not currently accessible.

Connecting Local Hardware to Complex Issues

To simulate differential equations on qubit-based quantum computers, discretise them. Researchers use finite difference to study first- and second-order differential operators. After discretisation, the finite-dimensional issue Hamiltonian converts the differential equation to quantum dynamics. Free particles follow the Schrödinger equation.

Problem Hamiltonian (up to negative sign) is a tridiagonal matrix with main diagonal components Hj,j = -2h-2 and sub/super diagonal elements Hj,j+1=Hj+1,j=h-2. Many differential equation Hamiltonians are sparse, banded matrices like tridiagonals.

A quantum computer that accurately simulates Hamiltonian time development without overhead is ideal. Current quantum computers use local spin operators (Pauli matrices) and allow only local qubit interactions. Qubit operator representations of sparse matrices may need considerable non-local interaction terms.

Researchers can simulate differential equations on quantum devices via Hamiltonian embedding. Translating the problem Hamiltonian to an embedding Hamiltonian (local spin operators) that physical hardware can better simulate is the main notion. The block-diagonal decomposition of a local spin operator embedding Hamiltonian H is H = diag(A,), where A is embedded in the upper left corner. By simulating H's temporal development on a quantum computer—presumably easier than simulating A—we implement A's evolution in the upper left corner e-iHt=diag(e-iAt,). The following example demonstrates.

Consider an 8-by-8 binary matrix A, with all zeros except for A1,8=A8,1=1 and Aj,j+1=Aj+1,j=1 for j = 1,…,7.

Circulant matrix is a simple Laplace operator with periodic boundary conditions. Three-qubit Hamiltonian A is used in quantum computing. A is basic, but block-encoding it as a quantum circuit is challenging and may take multiple qubits. Breaking A into fundamental Pauli strings produces various component operators, including 3-qubit interactions like XXX. Basic Hamiltonians are difficult for quantum computers to simulate.

As an alternative, Hamiltonian embedding represents A using simple quantum operations. Sx=X1+X2+X3+X4 is a 4-qubit Hamiltonian with Xj as a Pauli-X operator on site j. Hamiltonian Hilbert space has basis. Next, analyse Table 1's basis' 8-dimensional circulant unary code subspace:

The Hamming distance between adjacent codewords (including 1 and 8) is always 1, hence projecting Sx onto this subspace yields its target matrix A. By setting an initial state in this subspace and penalising leaking outside of it, A can be quantum mimicked without ancilla qubits using a few fundamental gates or analogue evolution time. Researchers disregard measurement findings outside the relevant subspace after selecting them. Other embedding procedures can fix this with antiferromagnetic or one-hot codes. For Hamiltonian embedding details, see the original work.

Also see Double-Transmon Coupler Improves Superconducting Quantum.

Successful Amazon Braket Device Demos Jiaqi Leng, Joseph Li, and Xiaodi Wu demonstrated Hamiltonian embedding's effectiveness on Amazon Braket's IonQ and QuEra quantum computing architectures.

Spatial discretisation was employed to model a two-dimensional Schrödinger equation's dynamics, generating a N2-sized problem Hamiltonian. To complement the QuEra device's Rydberg Hamiltonian (HRyd) architecture with positive Rydberg interaction coefficients (Vij), an antiferromagnetic embedding technique was devised.

This encoding method translates the problem Hamiltonian Hprob to the machine Hamiltonian HRyd by carefully selecting parameters and atom locations. The experiment employed 12 qubits in two chains (one for each spatial variable, x and y) with a discretisation number N=7. Despite hardware noise, experimental results matched numerical simulations qualitatively.

IonQ 1D Bosonic Dynamics Simulation

A one-dimensional Schrödinger equation for a single bosonic mode was simulated using IonQ's 25-qubit trapped ion quantum computer in another experiment.

In this case, Fock space truncation transferred the physical Hamiltonian to a finite-dimensional tridiagonal matrix. One-hot embedding applied this to IonQ. This embedding Hamiltonian featured up to 2-body interaction terms, making it suitable for near-term devices.

The Hamiltonian's “diagonal part” was handled by parameterised single-qubit Z rotations, while the “off-diagonal” part was implemented by the IonQ native Mølmer–Sørensen gate The Hamiltonian embedding framework can simulate a d-dimensional Schrödinger equation using O(d) qubits and polynomial interaction terms in d, giving an exponential quantum advantage over mesh-based PDE simulation.

Although noisy gates caused errors, the trials accurately recorded the position and kinetic energy operators' oscillating behaviour, matching theoretical closed-form solutions.

You may read IonQ in DARPA Quantum Computing QBI Program Stage B.