A Novel Quantum Method for Simulating High-Dimensional Dynamics on Near-Term Hardware: “Hamiltonian Embedding“
In an effort to close the gap between the constraints of existing near-term quantum hardware and complex computational challenges, a team of researchers from the University of Maryland and the University of California, Berkeley has developed a novel technique called Hamiltonian embedding. High-dimensional dynamics that are typically unsolvable by classical computers, such as those controlled by partial differential equations (PDEs), can now be simulated more easily with this technique.
PDE simulation is essential for scientists and engineers in a variety of fields, such as fluid dynamics, aircraft design, and the modelling of heat and sound propagation. However, classical skills are severely limited since the computing complexity for solving high-dimensional differential equations increases exponentially with the issue dimension. Although PDEs have been solved using quantum algorithms in the past, the majority of them require complex input models such as block-encoded matrices and Quantum Random Access Memory (QRAM), which calls for huge, fault-tolerant quantum computers that are not yet available.
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Connecting Local Hardware to Complicated Issues
Differential equations must be discretized to simulate them on qubit-based quantum computers. Here, researchers apply the finite difference approach to first and second-order differential operators. The issue Hamiltonian, a finite-dimensional Hamiltonian, maps the differential equation to a quantum dynamics problem after discretization. Free particles evolve according to the Schrödinger equation.
To discretize the (1D) Laplace operator, they use the stencil method:
The problem Hamiltonian (up to a minus 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. Often, problem Hamiltonians from differential equations are sparse, banded matrices like tridiagonal matrices.
A quantum computer that precisely mimics the problem Hamiltonian’s time development without overhead is ideal. However, contemporary quantum computers use local spin operators (Pauli matrices) and allow only local qubit interactions. Qubit operator representations of sparse matrices sometimes require substantially non-local interaction terms.
Researchers can use Hamiltonian embedding to simulate differential equations on quantum hardware. The primary idea is to translate the issue Hamiltonian to an embedding Hamiltonian (local spin operators) that physical hardware can emulate better. To build an embedding Hamiltonian H made of local spin operators that admits a block-diagonal decomposition: H = diag(A,*), where A is embedded in the upper left corner. Thus, by simulating H’s temporal evolution 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 illustrates the concept.
Let us consider an 8-by-8 binary-valued matrix A with all zeros except for A1,8=A8,1=1 and Aj,j+1=Aj+1,j=1 for j = 1,…,7.
This circulant matrix is a basic Laplace operator with periodic boundary conditions. In quantum computing, A is a three-qubit Hamiltonian. A has a basic structure, but expressing it as a quantum circuit (using block-encoding) is difficult and may require several qubits. Breaking A into fundamental Pauli strings yields several component operators, some of which, like XXX, involve 3-qubit interactions. Today’s quantum computers struggle to simulate a basic Hamiltonian.
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As an alternative, Hamiltonian embedding lets us represent A with simple quantum operations. Consider a 4-qubit Hamiltonian Sx=X1+X2+X3+X4, where Xj is a Pauli-X operator on site j. That Hamiltonian’s Hilbert space has basis. Next, analyze an 8-dimensional subspace spanned by Table 1’s basis (called circulant unary code:
| Circulant unary code (N = 8) | |||
|---|---|---|---|
| Basis Index | Codeword | Basis Index | Codeword |
| 1 | 0000 | 5 | 1111 |
| 2 | 0001 | 6 | 1110 |
| 3 | 0011 | 7 | 1100 |
| 4 | 0111 | 8 | 1000 |
Since the Hamming distance between adjacent codewords (including 1 and 8) is always 1, projecting Sx onto this subspace produces its target matrix A. A can be quantum simulated without ancilla qubits using a few elementary gates (or analogue evolution time) by establishing an initial state within this subspace and penalizing leaking outside of it. Researchers ignore measurement results outside the relevant subspace after selecting them. Using antiferromagnetic or one-hot codes, other embedding strategies can solve this problem. Read the original work for additional information on Hamiltonian embedding.
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Effective Amazon Braket Device Demonstrations
Jiaqi Leng, Joseph Li, and Xiaodi Wu were among the researchers who showed off the effectiveness of Hamiltonian embedding on two different quantum computing architectures that are available via Amazon Braket: IonQ and QuEra devices.
Using QuEra to Simulate 2D Schrödinger Dynamics
The time evolution of a quantum system in real space is described by the Schrödinger equation:
Spatial discretisation was used to simulate the dynamics produced by a two-dimensional Schrödinger equation, yielding a problem Hamiltonian of size N2. An antiferromagnetic embedding strategy was developed to complement the QuEra device’s architecture, which employs the Rydberg Hamiltonian (HRyd) featuring positive Rydberg interaction coefficients (Vij).
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By carefully choosing the parameters and atom positions, this encoding approach converts the problem Hamiltonian Hprob to the machine Hamiltonian HRyd. The experiment used a total of 12 qubits organized into two chains (one for each spatial variable, x and y), with a discretization number N=7. Despite noise caused by hardware, the experimental results qualitatively matched numerical simulations.
| Antiferromagnetic code (N = 7) | |||
|---|---|---|---|
| Basis Index | Codeword | Basis Index | Codeword |
| 1 | 010101 | 5 | 011010 |
| 2 | 010100 | 6 | 001010 |
| 3 | 010110 | 7 | 101010 |
| 4 | 010010 | ||
1D Bosonic Dynamics Simulation on IonQ
In a different experiment, IonQ’s 25-qubit trapped ion quantum computer was used to simulate a one-dimensional Schrödinger equation that corresponds to a single bosonic mode.
In this instance, Fock space truncation was used to transfer the physical Hamiltonian to a finite-dimensional tridiagonal matrix. The one-hot embedding was used to apply this to IonQ. This embedding produced an embedding Hamiltonian that was appropriate for near-term devices because it had up to 2-body interaction terms.
The “diagonal part” of the Hamiltonian was handled using parameterized single-qubit Z rotations, while the “off-diagonal” part was implemented using the IonQ native Mølmer–Sørensen gate. Importantly, a d-dimensional Schrödinger equation can be simulated with just O(d) qubits and polynomial interaction terms in d using the Hamiltonian embedding framework, providing an exponential quantum advantage over conventional mesh-based PDE simulation techniques.
The trials effectively captured the anticipated oscillating behaviour of the position and kinetic energy operators, matching theoretical closed-form solutions, despite significant error from noisy gates.
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Outlook
The range of quantum applications that are possible on modern quantum devices is expanded with the advent of Hamiltonian embedding. As a vital first step towards modelling high-dimensional differential equations, researchers anticipate that this method, which works with both analogue and digital quantum computers, will speed up the practical applications of quantum computing in scientific fields and beyond.
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