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Quasiprobabilistic Imaginary-Time Evolution on Quantum Computers Cover

Quasiprobabilistic Imaginary-Time Evolution on Quantum Computers

Quantum Information & Computation
Volume 26 (2026): Issue 1 (March 2026)
By: ,  ,  ,   and    
Open Access
|Jun 2026
Figures & tables

Figures & Tables

Figure 1.

Thermal expectation value estimation using ITE. For each data point, we average the error in the thermal expectation value |〈ψi|Oj|ψi〉 – tr(|ψβ〉〈ψβ|Oj)| over 30 randomly chosen Pauli observables Oj and 10 random TPQ states |ψi〉 = e−0.02HUi|0〉⊗n, where H is the 1D Heisenberg Hamiltonian on n qubits and Ui is a randomly chosen Clifford unitary. The blue and orange points show the values for exact and simulated ITE, respectively, where the simulation implements Algorithm 1.

Figure 2.

A schematic of a linear map 𝒯 implemented by sampling from its quasiprobability distribution (see Eq. (4), Algorithm 1). Given a quasiprobability decomposition of the map 𝒯, circuits are randomly sampled according to the distribution |qi|/γ, and their measurement outcomes are weighted by the factors sgn(qi)γ.

Figure 3.

Example of an operation 𝒯′ = 𝒯1 𝒯2 𝒯3 that we implement. Here 𝒯1, 𝒯2, 𝒯3 are 2-qubit operations applied on nearest-neighbor qubits at different locations. In the analysis of Lemma 2, we assume for simplicity that 𝒯1 = 𝒯2 = 𝒯3 = 𝒯.

Figure 4.

Scaling of the QPD cost γ with inverse temperature β for the decomposition of e−βH using different basis sets, where H is the 2-qubit Heisenberg Hamiltonian H = –XX – γγ – ZZ + II. The Takagi basis (orange) outperforms the EBL basis (blue), illustrating the utility of including entangling gates in the QPD basis set. The lower bound, obtained using the diamond norm [62,63], is shown in teal. (See Appendix C for the analysis of optimal sampling cost and the effect of adding identity to the Hamiltonian for ITE.)

Figure 5.

Energy estimation using ITE. We estimate the energy 〈ψt|H|ψt〉/〈ψt|ψt〉, where H is the 2-qubit Heisenberg Hamiltonian and |ψt〉 = (e−0.01H)t|00〉 is the imaginary-time evolved state with t Trotter steps. The blue, orange, and teal points show the values for exact, simulated, and hardware ITE, respectively, where the simulation (resp., demonstration) implements Algorithm 2 on classical (resp., quantum) hardware.

EBL basis [43]_ A complete basis for single-qubit linear maps that contains 10 trace-preserving elements ([1] to [RXY]) and 6 non-trace-preserving elements ([πX] to [πXY]), where we use the notation [U](·) := U(·)U†_ The map [P0] refers to projection onto the state |0〉_

[1]
[σX] =[H][S]2[H]
[σY] =[H][S]2[H][S]2
[σZ] =[S]2
[RX]=[ 1+iσX2]\left[ {{{1 + {\rm{i}}{\sigma ^X}} \over {\sqrt 2 }}} \right]=[H][S]3[H]
[RY]=[1+iσY2]\left[ {{{1 + {\rm{i}}{\sigma ^Y}} \over {\sqrt 2 }}} \right]=[S][H][S]3[H][S]3
[RZ]=[1+iσZ2]\left[ {{{1 + {\rm{i}}{\sigma ^Z}} \over {\sqrt 2 }}} \right]=[S]3
[RYZ]=[σY+σZ2]\left[ {{{{\sigma ^Y} + {\sigma ^Z}} \over {\sqrt 2 }}} \right]=[H][S]3[H][S]2
[RZX]=[σZ+σX2]\left[ {{{{\sigma ^Z} + {\sigma ^X}} \over {\sqrt 2 }}} \right]=[S]3[H][S]3[H][S]3
[RXY]=[σX+σY2]\left[ {{{{\sigma ^X} + {\sigma ^Y}} \over {\sqrt 2 }}} \right]=[H][S]2[H][S]3
[πX]=[1+σX2]\left[ {{{1 + {\sigma ^X}} \over 2}} \right]=[S][H][S][H][P0][H][S]3[H][S]3
[πY]=[1+σY2]\left[ {{{1 + {\sigma ^Y}} \over 2}} \right]=[H][S]3[H][P0][H][S][H]
[πZ]=[1+σZ2]\left[ {{{1 + {\sigma ^Z}} \over 2}} \right]=[P0]
[πYZ]=[σY+iσZ2]\left[ {{{{\sigma ^Y} + {\rm{i}}{\sigma ^Z}} \over 2}} \right]=[S][H][S][H][P0][H][S][H][S]3
[πZX]=[σZ+iσX2]\left[ {{{{\sigma ^Z} + {\rm{i}}{\sigma ^X}} \over 2}} \right]=[H][S]3[H][P0][H][S][H][S]2
[πXY]=[σX+iσY2]\left[ {{{{\sigma ^X} + {\rm{i}}{\sigma ^Y}} \over 2}} \right]=[P0][H][S]2[H]

Takagi Basis [64]_ A complete basis for 2-qubit CPTP maps_ {Bi}i=113\left\{ {{{\cal B}_i}} \right\}_{i = 1}^{13} are the first 13 elements of the EBL basis (A1)_ 𝒞𝒳, 𝒞𝒮, 𝒞ℋ, 𝒞ℋ𝒳 are channel versions of CNOT, controlled-phase, controlled-Hadamard, and NOT controlled with ±1 eigenstates of the Hadamard gate, respectively_ 𝒦i is the channel version of K := SH acting on qubit i_ SW and iSW are channel versions of the SWAP and iSWAP gates_ 𝒰 conjugated by 𝒱 denotes 𝒱† ⚬𝒰⚬𝒱 and “𝒰 + conjugation with 𝒦1/2, K1,2†{\cal K}_{1,2}^\dag ” collects the nine conjugations of 𝒰 by I12,K1,K2,K1†,K2†,K1⊗K2,K1⊗K2†,K1†⊗K2{I_{12}},{{\cal K}_1},{{\cal K}_2},{\cal K}_1^\dag ,{\cal K}_2^\dag ,{{\cal K}_1} \otimes {{\cal K}_2},{{\cal K}_1} \otimes {\cal K}_2^\dag ,{\cal K}_1^\dag \otimes {{\cal K}_2}, and K1†⊗K2†{\cal K}_1^ + \otimes {\cal K}_2^ + _

B1 – B169{Bi}i=113⊗{Bi}i=113\left\{ {{{\cal B}_i}} \right\}_{i = 1}^{13} \otimes \left\{ {{{\cal B}_i}} \right\}_{i = 1}^{13}
B170–B178𝒞𝒳 + conjugation with 𝒦1/2, K1,2†{\cal K}_{1,2}^\dag
B179–B187𝒳1 ⚬ 𝒞𝒳 ⚬ 𝒳1 + conjugation with 𝒦1/2, K1,2†{\cal K}_{1,2}^\dag
B188–B196𝒞𝒮 + conjugation with 𝒦1/2, K1,2†{\cal K}_{1,2}^\dag
B197–B205𝒞ℋ + conjugation with 𝒦1/2, K1,2†{\cal K}_{1,2}^\dag
B206–B214𝒞ℋ𝒳 + conjugation with 𝒦1/2, K1,2†{\cal K}_{1,2}^\dag
B215–B223𝒞ᒼ ⚬ ℋ1 + conjugation with 𝒦1/2, K1,2†{\cal K}_{1,2}^\dag
B224–B226𝒮W + conjugation with 𝒦2, K1,2†{\cal K}_{1,2}^\dag
B227–B232i𝒮W + conjugation with 𝒦1/2, K1,2†{\cal K}_{1,2}^\dag
B233–B241𝒮W ⚬ ℋ1 + conjugation with 𝒦1/2, K1,2†{\cal K}_{1,2}^\dag
DOI: https://doi.org/10.2478/qic-2026-0005 | Journal eISSN: 3106-0544 | Journal ISSN: 1533-7146
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Language: English
Page range: 89 - 113
Submitted on: Jun 2, 2025
Accepted on: Dec 25, 2025
Published on: Jun 4, 2026
Published by: Cerebration Science Publishing Co., Limited
In partnership with: Paradigm Publishing Services
Publication frequency: 1 issue per year
Keywords:
quantum algorithms,
imaginary-time evolution,
quantum error mitigation,
thermal expectation values
Related subjects:
Physics,
Quantum physics,
Mathematics,
Applied mathematics,
Computer sciences,
Computer sciences in the humanities,
Computer sciences,
Computer sciences in natural sciences

© 2026 Annie Ray, Esha Swaroop, Ningping Cao, Michael Vasmer, Anirban Chowdhury, published by Cerebration Science Publishing Co., Limited
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 26 (2026): Issue 1 (March 2026)
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