Selection and Improvement of Product Formulae for Best Performance of Quantum Simulation

Morales, Mauro E. S.; Costa, Pedro C. S.; Pantaleoni, Giacomo; Burgarth, Daniel K.; Sanders, Yuval R.; Berry, Dominic W.

doi:10.2478/qic-2025-0001

Figures & Tables

A histogram of the ratio of the error of the most accurate prior product formula SS8s19 to our new solution, in the case without processing. The histogram is over the 1024 samples of 64 × 64 matrices. The average improvement is about a factor of 100, with the lowest being about 35.

The spectral-norm error for simulating an 8-qubit transverse-field Ising Hamiltonian in Eq. (59) for our product formula Y8m10b (black), SS8s19 (blue), and SS8s21 (red) from Ref. [29]. The size of the time step is 1, and the components of the Hamiltonian are normalised to unit norm.

Our best-performing 8th order kernel and processor when setting m = 8_ The value of γ10 for the processor is chosen such that the sum is zero_

	Best 8th order kernel	Processor for kernel

w₁	0.21784176681731006074681969186513	−0.44324901019570126590495430949294	γ₁
w₂	0.1947017706053903224022456342907	0.25459857192003772850622377066944	γ₂
w₃	0.18372413281145589944261642180363	−0.73862036266779261573694538099739	γ₃
w₄	−0.37307499512657736825709230652023	−0.00024139614958652134370419495289618	γ₄
w₅	0.15757644257569146373033662060461	0.73873460354125365739379753874964	γ₅
w₆	−0.33342207567391682979227850551172	−0.20285971152536085519251666906017	γ₆
w₇	0.51788649682987924281787142226803	0.44989521689676869571827637424046	γ₇
w₈	0.21456475499897766986381219621761	0.29538398007876871184026747505657	γ₈
		−0.3364996155865700091428329802017	γ₉

The 10th order solution with lowest eigenvalue error in our numerical search_

	10th order solution with m = 17

w₁	−0.28371232689144296279654621726493
w₂	0.046779504778147381605331000278223
w₃	0.36845892382797770619657504217539
w₄	0.19186204094674514739760408197461
w₅	−0.53123134392680669702873064192428
w₆	−0.0081253242720827266680816105600661
w₇	−0.16389450414378567860032917538393
w₈	0.18514766119291405032528647881
w₉	0.5383584694754681989174668806505
w₁₀	−0.30583981835573485697292316732177
w₁₁	0.43199935609523301289295473774488
w₁₂	0.1510502301631786853020124612813
w₁₃	−0.35051099204829676098801520498121
w₁₄	0.1032971125844291674511513007661
w₁₅	0.15043936943817152697371946806229
w₁₆	0.12118469498650736511410491586846
w₁₇	0.10437742779547826358296681557444

The list of 6th order product formulae with their average errors_ The column labelled S2 indicates whether it is a product of S2_ The best result is highlighted in green, and is from 2006 [31]_

label	M	S₂	processing	reference	χ	Mχ^1/k	ζ	Mζ^1/k

S6m1	9	Y	N	first fractal product [2]	4.5 × 10⁻²	5.36	2.3 × 10⁻²	4.81
S6m2	25	Y	N	second fractal product [4]	1.3 × 10⁻⁵	3.83	2.0 × 10⁻⁷	1.91
Y6m3a	7	Y	N	Solution A in Table 1 of [3]	2.0 × 10⁻³	2.49	1.2 × 10⁻³	2.28
KL6s9a	9	Y	N	s9odr6a in Appendix A of [25]	2.9 × 10⁻⁴	2.31	1.7 × 10⁻⁴	2.12
KL6s9b	9	Y	N	s9odr6b in Appendix A of [25]	2.9 × 10⁻⁴	2.31	1.7 × 10⁻⁴	2.11
SS6s11	11	Y	N	Section 4.2 of [29]	4.2 × 10⁻⁵	2.05	1.4 × 10⁻⁵	1.71
SS6s13	13	Y	N	Section 4.2 of [29]	2.1 × 10⁻⁵	2.16	3.4 × 10⁻⁶	1.59
BM6M10	10	N	N	S₁₀ Table 2 of [47]	6.0 × 10⁻⁶	1.35	1.4 × 10⁻⁶	1.06
PPBCM6m9	9	N	Y	P₉6 Table 5 of [31]	5.0 × 10⁻⁶	1.18	4.3 × 10⁻⁷	0.78
PPBCM6m5	11	Y	Y	P₁₁6 in Table 6 of [31]	1.9 × 10⁻⁶	1.22	9.0 × 10⁻⁷	1.08
PPBCM6m6	13	Y	Y	P₁₃6 in Table 6 of [31]	4.8 × 10⁻⁷	1.15	2.5 × 10⁻⁷	1.03
BCE6m5	10	Y	Y	ψ5[6] \psi _5^{[6]} Table 8 of [40]	5.8 × 10⁻³	4.24	3.2 × 10⁻³	3.84
BCE6m6	12	Y	Y	ψ6[6] \psi _6^{[6]} Table 8 of [40]	7.7 × 10⁻⁵	2.47	2.0 × 10⁻⁵	1.98
BCE6m7	14	Y	Y	ψ7[6] \psi _7^{[6]} Table 8 of [40]	1.6 × 10⁻⁵	2.22	2.4 × 10⁻⁶	1.62
BCE6m8	16	Y	Y	ψ8[6] \psi _8^{[6]} Table 8 of [40]	4.6 × 10⁻⁶	2.06	3.2 × 10⁻⁷	1.32
BCE6m9	18	Y	Y	ψ9[6] \psi _9^{[6]} Table 8 of [40]	7.3 × 10⁻⁶	2.51	1.3 × 10⁻⁷	1.28
BCE6m10	20	Y	Y	ψ10[6] \psi _{10}^{[6]} Table 8 of [40]	3.2 × 10⁻⁶	2.42	9.9 × 10⁻⁹	0.93
BCE6m11	22	Y	Y	ψ11[6] \psi _{11}^{[6]} Table 8 of [40]	7.9 × 10⁻⁶	3.11	1.4 × 10⁻⁸	1.08

Comparison of constant factors ω for a selection of the lowest-error product formulae for 8th order_ We generate 1, 000 random Hamiltonians with d = 6 orbitals as in Eq_ (60) and compute the average ω_

label	M	processing	reference	ω	Mω^1/k

SS8s19	19	N	Section 4.3 of [29]	3.5 × 10⁻¹¹	0.94
PP8s13	13	Y	P₁₃8 in Table 6 of [31]	4.4 × 10⁻¹⁰	0.88
Y8m10	21	N	Table 1 (our new result)	8.5 × 10⁻¹²	0.87
Y8m10b	21	N	Table 1 (our new result)	1.3 × 10⁻¹²	0.68
YP8m8	17	Y	Table 2 (our new result)	1.7 × 10⁻¹²	0.57

Comparison of constant factors ω for a selection of the lowest-error product formulae for 6th order_ We generate 1, 000 random Hamiltonians with d = 4 orbitals as in Eq_ (60) and compute the average ω_

label	M	processing	reference	ω	Mω^1/k

PPBCM6m9	9	Y	P₉6 Table 5 of [31]	2.9 × 10⁻⁹	0.34
PPBCM6m6 BCE6m10	13	Y	P₁₃6 in Table 6 of [31]	1.4 × 10⁻⁹	0.43
PPBCM6m9	20	Y	ψ10[6] \psi _{10}^{[6]} Table 8 of [40]	3.3 × 10⁻¹¹	0.36

Comparison of constant factors ω for a selection of the lowest-error product formulae for 10th order_ We generate 1, 000 random Hamiltonians with d = 4 orbitals as in Eq_ (60) and compute the average ω_

label	M	processing	reference	ω	Mω^1/k

SS10s35	35	N	Section 4.4 of [29]	3.0 × 10⁻¹⁵	1.24
PP10s23	23	Y	P₂₃10 in Table 6 of [31]	1.5 × 10⁻¹²	1.51
Y10m17	35	N	Table 3	2.5 × 10⁻¹⁴	1.53

The list of 8th order product formulae with their average errors_ This table shows our new results for product formulae that improve over those in the prior literature_ The result that provides the most efficient simulations is highlighted in (dark) green, and is our processed product formula_ That provides the best performance due to the shorter length, but the lowest error is provided by our product formula highlighted in (dark) blue_ The lowest error results from prior work are highlighted in light blue (for the non-processed case), and light green (for the processed case)_ The processed product formula PP8s13 (P138) has larger error than the highlighted PP8s19, but better performance due to its shorter length_

label	M	processing	reference	χ	Mχ^1/k	ζ	Mζ^1/k

S8m1	27	N	first fractal product [2]	5.6 × 10⁻²	18.8	1.6 × 10⁻²	16.2
S8m2	125	N	second fractal product [4]	6.7 × 10⁻⁹	11.9	5.2 × 10⁻¹³	3.64
Y8m7d	15	N	Solution D in Table 2 of [3]	1.1 × 10⁻³	6.41	1.3 × 10⁻⁴	4.89
MC8s15	15	N	Table 2 of [24]	6.5 × 10⁻⁶	3.37	2.0 × 10⁻⁶	2.90
MC8s17	17	N	Table 2 of [24]	7.9 × 10⁻⁷	2.94	3.3 × 10⁻⁷	2.63
KL8s17a	17	N	s17odr8a in [25]	6.1 × 10⁻⁷	2.84	2.7 × 10⁻⁷	2.56
KL8s17b	17	N	s17odr8b in [25]	5.9 × 10⁻⁷	2.83	2.5 × 10⁻⁷	2.54
SS8s19	19	N	Section 4.3 of [29]	1.8 × 10⁻⁷	2.72	5.3 × 10⁻⁸	2.34
SS8s21	21	N	Section 4.3 of [29]	3.4 × 10⁻⁷	3.27	7.5 × 10⁻⁸	2.70
PP8s13	13	Y	P₁₃8 in Table 6 of [31]	1.2 × 10⁻⁶	2.37	6.5 × 10⁻⁷	2.19
PP8s19	19	Y	P₁₉8 in Table 6 of [31]	NA	NA	2.4 × 10⁻⁷	2.83
Y8m10	21	N	Table 1 (our new result)	5.8 × 10⁻⁸	2.61	7.0 × 10⁻⁹	2.01
Y8m10b	21	N	Table 1 (our new result)	6.3 × 10⁻⁷	3.53	5.4 × 10⁻¹⁰	1.46
YP8m8	17	Y	Table 2 (our new result)	5.3 × 10⁻⁸	2.09	8.1 × 10⁻¹⁰	1.24

The list of 10th order product formulae with their average errors_ The best result is highlighted in green, and is the product formula from 2005 [29]_ That product formula provides exceptional performance for 10th order, being about 280 times better than all others from prior work, and half the error of our solution (bottom line)_

label	M	processing	reference	χ	Mχ^1/k	ζ	Mζ^1/k

S10m1	81	N	first fractal product [2]	9.0 × 10⁻²	63.7	2.7 × 10⁻³	44.8
S10m2	625	N	second fractal product [4]	4.1 × 10⁻¹³	36.0	6.1 × 10⁻¹⁹	9.44
KL10s31a	31	N	s31odr10a in Appendix A of [25]	8.7 × 10⁻⁶	9.67	5.8 × 10⁻⁶	9.28
KL10s31b	31	N	s31odr10b in Appendix A of [25]	8.4 × 10⁻⁵	12.1	4.1 × 10⁻⁵	11.3
Tsi10s33	33	N	Table II of [26]	3.3 × 10⁻⁶	9.33	6.4 × 10⁻⁷	7.93
SS10s31	31	N	Section 4.4 of [29]	4.2 × 10⁻⁸	5.66	2.4 × 10⁻⁸	5.36
SS10s33	33	N	Section 4.4 of [29]	1.4 × 10⁻⁸	5.42	8.8 × 10⁻⁹	5.17
SS10s35	35	N	Section 4.4 of [29]	1.0 × 10⁻⁹	4.41	3.1 × 10⁻¹¹	3.11
Alberdi31	31	N	Appendix A of [48]	1.5 × 10⁻⁷	6.44	1.0 × 10⁻⁷	6.20
Alberdi33	33	N	Appendix A of [48]	8.4 × 10⁻⁸	6.47	5.5 × 10⁻⁸	6.20
Alberdi35	35	N	Appendix A of [48]	1.5 × 10⁻⁸	5.79	9.5 × 10⁻⁹	5.52
PP10s19	19	Y	P₁₉10 in Table 6 of [31]	NA	NA	6.2 × 10⁻⁶	5.73
PP10s23	23	Y	P₂₃10 in Table 6 of [31]	1.4 × 10⁻⁶	5.96	1.6 × 10⁻⁸	3.82
Y10m17	35	N	Table 3	1.9 × 10⁻⁸	5.91	6.1 × 10⁻¹¹	3.33

The list of 4th order product formulae_ The column labelled “processing” indicates whether the formula uses processor_ The label is the name we will refer to the formula by_ The constant factor in the error is denoted χ for spectral-norm error and ζ for eigenvalue error, and the corresponding quantities Mχ1/k and Mζ1/k are given_ The column labelled S2 indicates whether the product formula is a product of S2_ The best result is highlighted in green, and is from 2002 [47]_ The 2006 result PPBCM4m6 is slightly higher error, but it is not evident to 2 significant figures_

label	M	S₂	processing	reference	χ	Mχ^1/k	ζ	Mζ^1/k

S4m1	3	Y	N	first fractal product [2]	4.9 × 10⁻²	1.41	2.3 × 10⁻²	1.17
S4m2	5	Y	N	second fractal product [4]	3.0 × 10⁻³	1.17	3.3 × 10⁻⁴	0.67
O4M5	5	N	N	Ostmeyer Eq. (40) of [46]	3.0 × 10⁻⁴	0.66	7.7 × 10⁻⁵	0.47
BM4M6	6	N	N	S₆ Table 2 of [47]	1.6 × 10⁻⁴	0.67	2.4 × 10⁻⁵	0.42
PPBCM4m6	6	N	Y	P₆4 Table 5 of [31]	5.7 × 10⁻⁵	0.52	2.4 × 10⁻⁵	0.42
BCE4m3	6	Y	Y	ψ3[4] \psi _3^{[4]} Table 6 of [40]	1.6 × 10⁻²	2.15	1.5 × 10⁻³	1.17
BCE4m4	8	Y	Y	ψ4[4] \psi _4^{[4]} Table 6 of [40]	3.4 × 10⁻⁴	1.09	9.8 × 10⁻⁵	0.80
BCE4m5	10	Y	Y	ψ5[4] \psi _5^{[4]} Table 6 of [40]	6.7 × 10⁻⁵	0.90	2.1 × 10⁻⁵	0.68
BCE4m6	12	Y	Y	ψ6[4] \psi _6^{[4]} Table 6 of [40]	2.6 × 10⁻⁵	0.85	7.0 × 10⁻⁶	0.62
BCE4m7	14	Y	Y	ψ7[4] \psi _7^{[4]} Table 6 of [40]	1.3 × 10⁻⁵	0.85	3.0 × 10⁻⁶	0.58
BCE4m8	16	Y	Y	ψ8[4] \psi _8^{[4]} Table 6 of [40]	7.9 × 10⁻⁵	0.85	1.5 × 10⁻⁶	0.56
BCE4m9	18	Y	Y	ψ9[4] \psi _9^{[4]} Table 6 of [40]	5.3 × 10⁻⁶	0.86	8.6 × 10⁻⁷	0.55

Our best-performing 8th order solutions when setting m = 10_

	Best 8th order for spectral-norm error	Best 8th order for eigenvalue error

w₁	0.59358060400850625863514059265224	0.10467636532245895252340732579853
w₂	−0.46916012347004197296293264921328	−0.57896999331780988041471955125778
w₃	0.2743566425898467907228242878146	0.57503350160061785946141563279891
w₄	0.17193879484656773059919074965377	0.12231011868707029786561397542663
w₅	0.23439874482541384415430578747541	0.27793149999039524816733903301747
w₆	−0.48616424480326193899617759997914	−0.37349605088056728482635987352576
w₇	0.49617367388114660354871757044906	0.11575566589480463220616543972403
w₈	−0.32660218948439130114501815323814	0.1464645610975800618712569230326
w₉	0.23271679349369857679445410270557	−0.39443578322284085764474498594073
w₁₀	0.098249557414708533273471906180643	0.44370228726021218923197141183196

Comparison of constant factors ω for a selection of the lowest-error best product formulae for 6th order_ We generate 1, 000 random Hamiltonians with d = 6 orbitals as in Eq_ (60) and compute the average ω_

label	M	processing	reference	ω	Mω^1/k

PPBCM6m9	9	Y	P₉6 Table 5 of [31]	2.8 × 10⁻⁹	0.33
PPBCM6m6	13	Y	P₁₃6 in Table 6 of [31]	1.2 × 10⁻⁹	0.42
BCE6m10	20	Y	ψ10[6] \psi _{10}^{[6]} Table 8 of [40]	3.4 × 10⁻¹¹	0.36

Comparison of constant factors ω for a selection of the lowest-error product formulae for 10th order_ We generate 1, 000 random Hamiltonians with d = 6 orbitals as in Eq_ (60) and compute the average ω_

label	M	processing	reference	ω	Mω^1/k

SS10s35	35	N	Section 4.4 of [29]	3.0 × 10⁻¹⁵	1.24
PP10s23	23	Y	P₂₃10 in Table 6 of [31]	2.3 × 10⁻¹²	1.58
Y10m17	35	N	Table 3	2.6 × 10⁻¹⁴	1.53

The 8th order kernel tailored for large time step size t = 1_82 to improve the threshold_

	kernel for large time step

w₁	0.1777372900430394
w₂	0.2862580532195395
w₃	0.1701306063199336
w₄	−0.3746748008394162
w₅	0.1485267804844835
w₆	−0.3773225725485588
w₇	0.5395886879620081
w₈	0.2210419534887659

Comparison of constant factors ω for a selection of the lowest-error product formulae for 8th order_ We generate 1, 000 random Hamiltonians with d = 4 orbitals as in Eq_ (60) and compute the average ω_

label	M	processing	reference	ω	Mω^1/k

SS8s19	19	N	Section 4.3 of [29]	3.4 × 10⁻¹¹	0.93
PP8s13	13	Y	P₁₃8 in Table 6 of [31]	4.2 × 10⁻¹⁰	0.87
Y8m10	21	N	Table 1 (our new result)	8.7 × 10⁻¹²	0.87
Y8m10b	21	N	Table 1 (our new result)	1.5 × 10⁻¹²	0.68
YP8m8	17	Y	Table 2 (our new result)	2.3 × 10⁻¹²	0.59

Selection and Improvement of Product Formulae for Best Performance of Quantum Simulation

Figures & Tables

Figure 1.

Figure 2.

Our best-performing 8th order kernel and processor when setting m = 8_ The value of γ10 for the processor is chosen such that the sum is zero_

The 10th order solution with lowest eigenvalue error in our numerical search_

The list of 6th order product formulae with their average errors_ The column labelled S2 indicates whether it is a product of S2_ The best result is highlighted in green, and is from 2006 [31]_

Comparison of constant factors ω for a selection of the lowest-error product formulae for 8th order_ We generate 1, 000 random Hamiltonians with d = 6 orbitals as in Eq_ (60) and compute the average ω_

Comparison of constant factors ω for a selection of the lowest-error product formulae for 6th order_ We generate 1, 000 random Hamiltonians with d = 4 orbitals as in Eq_ (60) and compute the average ω_

Comparison of constant factors ω for a selection of the lowest-error product formulae for 10th order_ We generate 1, 000 random Hamiltonians with d = 4 orbitals as in Eq_ (60) and compute the average ω_

Our best-performing 8th order solutions when setting m = 10_

Comparison of constant factors ω for a selection of the lowest-error best product formulae for 6th order_ We generate 1, 000 random Hamiltonians with d = 6 orbitals as in Eq_ (60) and compute the average ω_

Comparison of constant factors ω for a selection of the lowest-error product formulae for 10th order_ We generate 1, 000 random Hamiltonians with d = 6 orbitals as in Eq_ (60) and compute the average ω_

The 8th order kernel tailored for large time step size t = 1_82 to improve the threshold_

Comparison of constant factors ω for a selection of the lowest-error product formulae for 8th order_ We generate 1, 000 random Hamiltonians with d = 4 orbitals as in Eq_ (60) and compute the average ω_

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