We propose a simple scheme to estimate fermionic observables and Hamiltonians relevant in quantum chemistry and correlated fermionic systems. Our approach is based on implementing a measurement that jointly measures noisy versions of any product of two or four Majorana operators in an N mode fermionic system. To realize our measurement we use: (i) a randomization over a set of unitaries that realize products of Majorana fermion operators; (ii) a unitary, sampled at random from a constant-size set of suitably chosen fermionic Gaussian unitaries; (iii) a measurement of fermionic occupation numbers; (iv) suitable post-processing. Our scheme can estimate expectation values of all quadratic and quartic Majorana monomials to ϵ precision using <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41534_2025_957_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="116" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal{O}}(N\log (N)/{\epsilon }^{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mrow> <mo>(</mo> <mrow> <mi>N</mi> <mi>log</mi> <mrow> <mo>(</mo> <mrow> <mi>N</mi> </mrow> <mo>)</mo> </mrow> <mo>/</mo> <msup> <mrow> <mi>ϵ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41534_2025_957_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="122" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal{O}}({N}^{2}\log (N)/{\epsilon }^{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mi>N</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mi>log</mi> <mrow> <mo>(</mo> <mrow> <mi>N</mi> </mrow> <mo>)</mo> </mrow> <mo>/</mo> <msup> <mrow> <mi>ϵ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> measurement rounds respectively, matching the performance offered by fermionic classical shadows<CitationRef CitationID="CR1">1</CitationRef>,<CitationRef CitationID="CR2">2</CitationRef>. In certain settings, such as a rectangular lattice of qubits which encode an N mode fermionic system via the Jordan-Wigner transformation, our scheme can be implemented in circuit depth <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41534_2025_957_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal{O}}({N}^{1/2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mi>N</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41534_2025_957_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal{O}}({N}^{3/2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mi>N</mi> </mrow> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> two-qubit gates, offering an improvement over fermionic and matchgate classical shadows that require depth <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41534_2025_957_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal{O}}(N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mrow> <mo>(</mo> <mrow> <mi>N</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41534_2025_957_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal{O}}({N}^{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mi>N</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> two-qubit gates. By benchmarking our method on exemplary molecular Hamiltonians and observing performances comparable to fermionic classical shadows, we demonstrate a novel, competitive alternative to existing strategies.

A simple and efficient joint measurement strategy for estimating fermionic observables and Hamiltonians

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2025年4月7日

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Joanna Majsak,
Daniel McNulty,
Michał Oszmaniec

摘要

We propose a simple scheme to estimate fermionic observables and Hamiltonians relevant in quantum chemistry and correlated fermionic systems. Our approach is based on implementing a measurement that jointly measures noisy versions of any product of two or four Majorana operators in an N mode fermionic system. To realize our measurement we use: (i) a randomization over a set of unitaries that realize products of Majorana fermion operators; (ii) a unitary, sampled at random from a constant-size set of suitably chosen fermionic Gaussian unitaries; (iii) a measurement of fermionic occupation numbers; (iv) suitable post-processing. Our scheme can estimate expectation values of all quadratic and quartic Majorana monomials to ϵ precision using ${\mathcal{O}}(N\log (N)/{\epsilon }^{2})$ $O (N \log (N) / ϵ^{2})$ and ${\mathcal{O}}({N}^{2}\log (N)/{\epsilon }^{2})$ $O (N^{2} \log (N) / ϵ^{2})$ measurement rounds respectively, matching the performance offered by fermionic classical shadows^1,2. In certain settings, such as a rectangular lattice of qubits which encode an N mode fermionic system via the Jordan-Wigner transformation, our scheme can be implemented in circuit depth ${\mathcal{O}}({N}^{1/2})$ $O (N^{1 / 2})$ with ${\mathcal{O}}({N}^{3/2})$ $O (N^{3 / 2})$ two-qubit gates, offering an improvement over fermionic and matchgate classical shadows that require depth ${\mathcal{O}}(N)$ $O (N)$ and ${\mathcal{O}}({N}^{2})$ $O (N^{2})$ two-qubit gates. By benchmarking our method on exemplary molecular Hamiltonians and observing performances comparable to fermionic classical shadows, we demonstrate a novel, competitive alternative to existing strategies.