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

Paul K. Faehrmann, Jens Eisert, Richard Kueng

TL;DR

The paper tackles the challenge of extracting richer information from the Hadamard test in the NISQ to ISQ transition by coupling it with classical shadows on the $n$-qubit work register. The authors propose a hybrid framework where the Hadamard test's auxiliary-qubit measurements are augmented with system-register shadows, enabling concurrent estimation of fidelities to known eigenstates, the state energy $\mathrm{tr}(H\rho)$, and eigenstate content, while preserving existing trace-estimation goals. They demonstrate how post-measurement states $\rho(I)$ and $\rho(Z)$ can be shadowed to obtain these quantities with rigorous error guarantees, discuss the trade-offs between global and local shadows, and quantify sampling implications $\mathcal{O}(\epsilon^{-2})$ versus $\mathcal{O}(\epsilon^{-1})$ in certain regimes. Furthermore, they introduce anti-controlled unitaries to linearize Hadamard-test outputs, enabling spectral comparison and eigenstate discrimination via cosines of energy differences, and they outline how randomized ensembles can approximate time-evolution operators. Overall, this approach offers a path to richer state-characterization and spectral analysis on shallow quantum devices, potentially accelerating progress toward practical fault-tolerant quantum computation.

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.

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

TL;DR

The paper tackles the challenge of extracting richer information from the Hadamard test in the NISQ to ISQ transition by coupling it with classical shadows on the -qubit work register. The authors propose a hybrid framework where the Hadamard test's auxiliary-qubit measurements are augmented with system-register shadows, enabling concurrent estimation of fidelities to known eigenstates, the state energy , and eigenstate content, while preserving existing trace-estimation goals. They demonstrate how post-measurement states and can be shadowed to obtain these quantities with rigorous error guarantees, discuss the trade-offs between global and local shadows, and quantify sampling implications versus in certain regimes. Furthermore, they introduce anti-controlled unitaries to linearize Hadamard-test outputs, enabling spectral comparison and eigenstate discrimination via cosines of energy differences, and they outline how randomized ensembles can approximate time-evolution operators. Overall, this approach offers a path to richer state-characterization and spectral analysis on shallow quantum devices, potentially accelerating progress toward practical fault-tolerant quantum computation.

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 -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 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.

Paper Structure

This paper contains 14 sections, 1 theorem, 32 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Hadamard test and its uses
  3. The Hadamard test in the light of shadows
  4. Using system register measurements for fidelity estimation
  5. Using system register measurements for energy estimation
  6. Using system register measurements for purity and eigenstateness
  7. Adding anti-controlled unitaries to the Hadamard test
  8. Discussion and outlook
  9. The Hadamard test as known from the literature
  10. Considering post-measurement states
  11. Measurements on the system register
  12. Summary of quantities accessible with the Hadamard test
  13. Summary of quantities accessible with the extended Hadamard test
  14. Rigorous performance guarantees for classical shadows

Key Result

Theorem 1

Classical shadows of size $N$ suffice to predict $M$ arbitrary linear target functions $\mathrm{tr}\!\left(O_1\rho\right),\ldots,\mathrm{tr}\!\left(O_M\rho\right)$ up to additive error $\epsilon$ given that with probability at least $1-\delta$. The definition of the norm $\lVert O_i \rVert_\mathrm{shadow}$ depends on the ensemble of unitary transformations used to create the classical shadow.

Figures (3)

  • Figure 1: Cartoon illustration of the proposed adaptation of the Hadamard test: (left) The standard Hadamard test circuit allows for the estimation of $\mathrm{Re}(\mathrm{tr}\!\left(U\rho\right))$ or $\mathrm{Im}(\mathrm{tr}\!\left(U\rho\right))$ when using $\phi=0$ and $\phi=\pi/2$ respectively. Here, $V$ labels the state preparation unitary, but the state can also be mixed. (right) Instead of disregarding the system register, we can perform local or global shadow estimation of the post-measurement state by applying random local or global Clifford gates ($C$) and thereby extract so-far unused information.
  • Figure 2: The Hadamard test with an additional anti-controlled unitary $W$ can help in quantum dynamics and linear algebra by linearizing the output to allow for randomized approaches faehrmannRandomizingMultiproductFormulas2022wangQubitEfficientRandomizedQuantum2024, or to compare spectra of unitaries and determine eigenstateness of a state as discussed in the main text.
  • Figure 3: The standard Hadamard test.

Theorems & Definitions (1)

  • Theorem 1: Performance guarantees for classical shadows huangPredictingManyProperties2020c