In the shadow of the Hadamard test: Using the garbage state for good and further modifications

Abstract

The Hadamard test is naturally suited for the intermediate regime between the current era of noisy quantum devices and complete fault tolerance. Its applications use measurements of the auxiliary qubit to extract information, but disregard the system register completely. Separate advances in classical representations of quantum states via classical shadows allow the implementation of even global classical shadows with shallow circuits. This work combines the Hadamard test on a single auxiliary readout qubit with classical shadows on the remaining $n$-qubit work register. We argue that this combination inherits the best of both worlds and discuss statistical phase estimation as a vignette application. There, we can use the Hadamard test to estimate eigenvalues on the auxiliary qubit, while classical shadows on the remaining $n$ qubits provide access to additional features such as, (i) fidelity with certain pure quantum states, (ii) the initial state's energy and (iii) how pure and how close the initial state is to an eigenstate of the Hamiltonian. Finally, we also discuss how anti-controlled unitaries can further augment this framework.

6+3 pages, 3 figures

References in corpus (19)

• Predicting Many Properties of a Quantum System from Very Few Measurements
• Logical quantum processor based on reconfigurable atom arrays
• Quantum error correction below the surface code threshold
• The randomized measurement toolbox
• Heisenberg-limited ground state energy estimation for early fault-tolerant quantum computers
• Hardware-efficient quantum error correction via concatenated bosonic qubits
• A randomized quantum algorithm for statistical phase estimation
• Error mitigation via verified phase estimation
• Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision
• Shallow shadows: Expectation estimation using low-depth random Clifford circuits
• Quantum algorithms: A survey of applications and end-to-end complexities
• Randomizing multi-product formulas for Hamiltonian simulation
• Beyond NISQ: The Megaquop Machine
• Implementing any Linear Combination of Unitaries on Intermediate-term Quantum Computers
• Estimating expectation values using approximate quantum states
• Qubit-Efficient Randomized Quantum Algorithms for Linear Algebra
• Quantifying quantum coherence of multiple-charge states in tunable Josephson junctions
• Optimizing the information extracted by a single qubit measurement
• Algorithmic Shadow Spectroscopy

Cited by in corpus (4)

• Resource-Efficient Hadamard Test Tailored Variational Framework for Nonlinear Dynamics on Quantum Computers
• Experimental Quantum Channel Purification
• Deterministic Ground State Preparation via Power-Cosine Filtering of Time Evolution Operators
• Estimation of multivariate traces of states given partial classical information