1. Introduction
As quantum computing enters the noisy intermediate-scale quantum (NISQ) era, one of the key challenges is managing errors and limited coherence times. Quantum Subspace Expansion (QSE) is a variational error mitigation technique that improves the accuracy of quantum algorithms—especially those solving problems in quantum chemistry, materials science, and optimization.
QSE is often used alongside variational algorithms like the Variational Quantum Eigensolver (VQE). It allows researchers to access more accurate solutions by expanding the quantum system’s state into a subspace, capturing more physics than the original variational state alone.
2. Why Was QSE Introduced?
Quantum algorithms that rely on variational methods work by preparing an approximate quantum state on a real device, measuring observables, and adjusting parameters to find optimal values (e.g., the lowest energy state of a molecule).
But due to:
- Hardware noise,
- Limited depth of circuits, and
- Imperfect variational ansätze (trial states),
the solution is often far from exact.
QSE provides a way to improve upon the variational result by considering a subspace of related quantum states that contain more information about the system—without needing longer or more complex circuits.
3. The Core Idea Behind QSE
Imagine you’re trying to find the best solution to a problem (like the ground state energy of a molecule) using a trial quantum state. QSE assumes that the true solution might lie not exactly in the state you’ve prepared, but somewhere in a neighborhood of states around it.
Instead of relying only on the initial variational state, QSE constructs a subspace made from that state and a few others obtained by applying specific quantum operations. Then, the algorithm performs a classical computation using measurements from these expanded states to refine the result.
4. Step-by-Step Overview of How QSE Works
Step 1: Prepare the Ground State Ansatz
- Begin with a variational algorithm like VQE to generate a trial state.
- This state should approximate the ground state of the system of interest.
Step 2: Define a Set of Expansion Operators
- Choose a set of simple quantum operations (like single or two-qubit excitations) that you’ll apply to the initial state.
- These operators define a subspace of states close to your original ansatz.
Step 3: Generate the Subspace
- Apply the expansion operators to the initial state to create new basis states.
- Together, the original state and the expanded states form the quantum subspace.
Step 4: Measure the Subspace
- Use the quantum computer to measure overlaps and Hamiltonian expectation values between these new basis states.
- This requires running circuits for each pair of expanded states to gather measurement data.
Step 5: Build Matrices on Classical Hardware
- Construct a Hamiltonian matrix and an overlap matrix using the measurement results.
- These matrices represent the behavior of the system within the subspace.
Step 6: Solve the Generalized Eigenvalue Problem
- Classically compute the eigenvalues of the matrices to find improved energy estimates.
- The lowest eigenvalue gives a better approximation of the ground state energy than the original VQE alone.
Step 7: Analyze Excited States (Optional)
- QSE also allows extraction of excited state energies from the same data by analyzing higher eigenvalues—something standard VQE doesn’t provide.
5. Benefits of Quantum Subspace Expansion
Improved Accuracy
QSE provides a systematic way to refine results without significantly increasing quantum circuit complexity.
Error Mitigation
It helps correct errors introduced by decoherence, gate noise, or shallow circuit depth—key limitations in current quantum systems.
Access to Excited States
Unlike standard VQE, QSE enables access to excited states, which are important in chemistry and materials modeling.
Hybrid Efficiency
QSE fits well in the quantum-classical hybrid paradigm, where the quantum device handles state preparation and measurements, and classical computers solve matrix equations.
6. Real-World Applications of QSE
Quantum Chemistry
QSE is particularly useful for simulating molecular systems where accurate energies are needed for ground and excited states. Examples include hydrogen molecules, lithium hydride, and small organic compounds.
Materials Science
QSE can improve simulations of solid-state systems where energy precision matters for understanding conductivity, magnetism, and phase transitions.
Noise-Resilient Simulations
In NISQ environments, where hardware errors are unavoidable, QSE can significantly improve the fidelity of results from quantum simulations.
Molecular Excitation Studies
For understanding how molecules absorb energy or react chemically, knowing their excited states is crucial—QSE makes this possible with fewer resources.
7. Challenges and Limitations
Measurement Overhead
The number of measurements required increases with the number of expansion operators, making QSE resource-intensive for large subspaces.
Operator Selection
The quality of results depends heavily on which expansion operators are chosen. Too many can introduce noise; too few might miss important physics.
Classical Bottlenecks
Although QSE shifts complexity to the classical part of the workflow, solving large eigenvalue problems can become expensive for big systems.
Hardware Calibration
Accurate measurements of overlaps and expectation values demand well-calibrated, low-noise hardware.
8. Recent Advances and Extensions
- Adaptive QSE: Dynamically chooses the most relevant expansion operators to reduce measurement costs.
- Symmetry-aware QSE: Leverages system symmetries to reduce the subspace size.
- Integration with Error Mitigation: Combines QSE with error mitigation techniques like zero-noise extrapolation.
- Use in Near-Term Devices: Companies like IBM and startups like Zapata and QC Ware are experimenting with QSE in cloud-based quantum systems.
9. Future Outlook
QSE continues to gain relevance as quantum devices improve in qubit count and fidelity. It offers a low-cost enhancement for algorithms that are already practical in NISQ systems, like VQE and QAOA (Quantum Approximate Optimization Algorithm). As part of the growing toolbox of hybrid quantum-classical techniques, QSE will likely play a central role in:
- Developing more accurate quantum simulations,
- Supporting quantum chemistry research,
- And enabling meaningful use of early quantum computers.