Tensor Networks Quantum Computing
A Novel Algorithm Integrates Quantum and Classical Computing to Improve the Accuracy of Quantum Simulations
To simulate quantum many-body systems more accurately than was previously achievable with either technique alone, researchers from IBM Quantum, IBM Research Europe, IBM Research Almaden, IBM Thomas J Watson Research Centre, Donostia International Physics Centre (DIPC), and the University of Central Florida have developed a novel algorithm that works in concert with tensor networks and quantum computation. This novel technique, which is described in the publication “Tensor Network enhanced Dynamic Multiproduct Formulas,” overcomes significant drawbacks that both classical and quantum simulators must contend with.
Understanding out-of-equilibrium features in disciplines like chemistry, material science, and high-energy physics requires the simulation of Hamiltonian dynamics. However, simulations are limited to tiny systems or short evolution durations because of the exponential cost of representing entangled quantum states, which is a challenge for conventional computers. Theoretically, quantum computers may effectively model these dynamics, but because present processors are not fault-tolerant, errors can accumulate and high-precision simulations on near-term devices become unaffordable.
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The Hybrid Benefit: Multi-Product Formulas and Tensor Networks
The new algorithm suggests a hybrid solution and is led by Niall F. Robertson, Bibek Pokharel, and Sergiy Zhuk, among others. It makes use of:
- Multiproduct Formulas (MPF): This method significantly reduces algorithmic error by linearly combining multiple Trotter product formula approximations of time evolution operators. This dynamic MPF approach, in contrast to conventional static MPFs, employs time-dependent coefficients that are optimized to minimize error, thereby enhancing numerical stability and accuracy over extended simulation times.
- Matrix Product Operators (MPO): To effectively compute the coefficients required for the MPF, the classical part of the algorithm makes use of MPOs. One important finding is that even in cases when the time-evolved quantum states are highly entangled, several overlap quantities that are essential for MPF optimization may be expressed as weakly entangling operators. In contrast to directly storing quantum states (logarithm of bond dimension scales proportionally to T), this enables the classical computation of these coefficients with favourable memory scaling (logarithm of bond dimension scales proportionally to T times dt squared, where T is time and dt is Trotter step). This indicates that, compared to conventional classical simulations for quantum states, the algorithm’s classical processing stages require less memory.
- AQCtensor Algorithm: The Approximate Quantum Compiling (AQCtensor) algorithm is also incorporated into the workflow. It finds short-depth quantum circuits that approximate initial time evolution segments (up to time t1) using tensor networks. For longer simulation times (t2), this optimized circuit is then coupled with extra Trotter steps, which successfully lowers the total depth of the quantum circuits needed and, consequently, reduces device errors.
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Demonstrated on IBM Quantum Computers
Using two IBM quantum computers, ibm_torino and ibm_kyiv, the entire process was illustrated on a one-dimensional quantum simulation issue with 50 qubits. In comparison to a single, deeper Trotter circuit implemented on quantum hardware, the trials demonstrated that the dynamic MPO-MPF method produced more accurate results. For example, compared to a pure Trotter simulation with χ=400 or a pure Matrix Product State simulation with χ=100, the MPO-MPF technique with a classical bond dimension of χ=50 achieved reduced inaccuracy.
Additionally, the hybrid technique enabled dependable simulations at longer overall evolution durations when paired with the AQCtensor algorithm. To reduce device errors on the quantum hardware, error suppression strategies such Pauli twirling, twirled readout extinction, and dynamical decoupling were used. In order to further improve outcomes by modelling and scaling noise, ibm_kyiv additionally used Probabilistic Error Amplification (PEA).
The substantial potential of combining quantum computation with traditional tensor network methods is demonstrated by this study. According to the results, this hybrid approach may offer a significant benefit in quantum simulation by lowering algorithmic and device mistakes and maybe paving the way for the future simulation of more intricate quantum systems than one-dimensional spin chains.
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In Conclusion
This study presents a hybrid quantum-classical method for modelling quantum Hamiltonian dynamics, with an emphasis on out-of-equilibrium characteristics that are pertinent to material science and chemistry. In order to lower the computing cost of simulation, the fundamental methodology combines quantum Multi Product Formulas (MPF) with conventional tensor network methods. The use of Matrix Product Operators (MPO) for the classical calculation of dynamic coefficients is a significant invention.
For some simulation situations, MPO shows better memory scaling than traditional Matrix Product States (MPS). The effectiveness of this MPO-MPF approach in conjunction with AQCtensor, an approximation quantum compiler, on real quantum hardware is also examined in the research. It demonstrates enhanced accuracy and longer simulation periods by reducing device faults using strategies like Pauli twisting and dynamical decoupling.
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