The variational principle provides a powerful framework for certifying quantum solutions, particularly in optimization and ground-state energy estimation. This pathway enables solution quality to be ranked and benchmarked, even when exact answers are unknown or classically intractable.
Classical Ranking Capability
Variational algorithms guarantee that the estimated energy is an upper bound to the true ground-state energy.
This allows for direct comparison between quantum and classical methods, even in the absence of exact solutions.
As noted in the paper, this property is central to establishing trust in variational quantum outputs.
"In variational approaches, we are always assured that the reported number is an upper bound to the true energy value."
Spectrum of Approaches
The variational landscape spans a wide range of algorithms, each with different trade-offs in depth, precision, and hardware requirements:
Quantum Phase Estimation (QPE)
Offers exponential precision scaling but requires fault-tolerant execution.
This method provides the highest theoretical precision but demands significant quantum error correction capabilities.
Variational Quantum Eigensolvers (VQE)
Suitable for near-term devices, though limited by statistical uncertainty and measurement overhead.
VQE represents a practical approach for current noisy intermediate-scale quantum (NISQ) devices.
Sample-based Quantum Diagonalization (SQD)
A hybrid method that projects quantum data into a classical subspace for high-precision diagonalization.
"SQD has already been used to estimate energies for systems beyond exact diagonalization... robust to quantum sampling errors."
Quantum-Classical Integration
Modern hybrid algorithms like SQD and Sample-based Krylov Quantum Diagonalization (SKQD) combine quantum sampling with classical diagonalization.
- These methods store quantum-derived subspaces in classical memory, enabling deterministic, high-precision verification
- They inherit convergence guarantees from QPE while avoiding the need for deep circuits or variational ansätze
"Both SQD and SKQD achieve the highest degree of verifiability... allowing for a high-precision ranking of their accuracy."
Why It Matters
Variational methods offer a scalable and certifiable route to quantum advantage. By leveraging the variational principle and integrating classical post-processing, these algorithms enable mathematically grounded benchmarking—even in the absence of full fault tolerance.