H₂ VQE|41 results

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H₂ Ground State Energy via VQE

Variational quantum chemistry on 2 qubits -- from simulation to hardware

We calculate the ground state energy of molecular hydrogen using a 2-qubit VQE on four quantum processors. Chemical accuracy achieved on IBM Torino (0.22 kcal/mol with TREX) and QI Tuna-9 (0.92 kcal/mol with hybrid PS+REM). IQM Garnet reaches 14.26 kcal/mol after fixing a platform-specific CNOT decomposition error. The emulator nails chemical accuracy across 14 bond distances.

Research Question

Can a 2-qubit variational quantum eigensolver (VQE) calculate the ground state energy of molecular hydrogen within chemical accuracy (1.6 milliHartree), and how does hardware noise affect the result?

Prior Work

The Variational Quantum Eigensolver (VQE) was proposed by Peruzzo et al. in 2014 as a hybrid quantum-classical algorithm for finding molecular ground states. It is considered one of the most promising near-term applications of quantum computing because it can tolerate some hardware noise through its variational nature.

For molecular hydrogen (H2), the electronic structure problem can be mapped to qubits using the Jordan-Wigner or Bravyi-Kitaev transformation. The minimal STO-3G basis set gives a 4-qubit Hamiltonian, but Bravyi-Kitaev transformation with qubit tapering (exploiting Z2 symmetries) reduces this to just 2 qubits -- the minimum for a non-trivial quantum chemistry calculation.

Sagastizabal et al. (2019) demonstrated VQE on a superconducting transmon processor, achieving chemical accuracy at the equilibrium bond distance. Our experiment replicates and extends this work across three platforms, including a full 14-point dissociation curve on the emulator.

The gold standard is chemical accuracy: getting within 1.6 milliHartree (1 kcal/mol) of the exact Full Configuration Interaction (FCI) energy. At the equilibrium bond distance of 0.735 Å, the FCI energy is -1.1373 Hartree.

Method

We use the 2-qubit Hamiltonian obtained by Bravyi-Kitaev transformation with qubit tapering of the H2 STO-3G Hamiltonian in the correct parity sector. The Hamiltonian has the form:

H = g0I + g1Z0 + g2Z1 + g3Z0Z1 + g4X0X1 + g5Y0Y1

Ansatz: RY(θ) on qubit 0, then CNOT(q[0], q[1]). The single parameter θ is optimized classically.

Measurement protocol: Three separate circuits measure in the Z, X, and Y bases to reconstruct all expectation values. 4096+ shots per basis per backend.

Dissociation curve: The emulator sweeps 14 bond distances from 0.3 to 3.0 Å with 65,536 shots each, using pre-computed optimal θ values at each distance.

Results

Platform Comparison

Backend	Type	Key Metric	Date
IBM Torino (133q)	Hardware	-1.1226 Ha (9.22 kcal/mol)	2/11/2026
IBM Torino (133q)	Hardware	-1.1197 Ha (11 kcal/mol)	2/10/2026
iqm-garnet	Emulator	-1.1146 Ha (14.26 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	--	2/10/2026
QI Emulator	Emulator	-1.1385 Ha (0.75 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.1310 Ha (3.94 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.1293 Ha (5.01 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.1358 Ha (0.92 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.1213 Ha (10.02 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.1329 Ha (2.79 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.1275 Ha (6.14 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.1263 Ha (6.93 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.1160 Ha (13.4 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.1291 Ha (5.13 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.1294 Ha (4.94 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.1213 Ha (10.02 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.1221 Ha (9.53 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.1238 Ha (8.5 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.1230 Ha (9 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.1272 Ha (6.31 kcal/mol)	2/10/2026
QI Emulator	Emulator	-1.1296 Ha (4.83 kcal/mol)	2/10/2026
QI Emulator	Emulator	-1.1415 Ha (2.64 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-0.9147 Ha (13.42 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.1325 Ha (3.04 kcal/mol)	2/10/2026
QI Emulator	Emulator	-0.9362 Ha (0.09 kcal/mol)	2/10/2026
QI Emulator	Emulator	-0.9454 Ha (2.05 kcal/mol)	2/10/2026
QI Emulator	Emulator	-0.9940 Ha (2.62 kcal/mol)	2/10/2026
QI Emulator	Emulator	-1.1034 Ha (1.4 kcal/mol)	2/10/2026
QI Emulator	Emulator	-1.1364 Ha (0.6 kcal/mol)	2/10/2026
QI Emulator	Emulator	-1.0583 Ha (2 kcal/mol)	2/10/2026
iqm-garnet	Emulator	-0.3105 Ha (518.85 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-0.9210 Ha (17.32 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-0.9779 Ha (12.68 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.0946 Ha (4.12 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.0393 Ha (9.98 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.1222 Ha (9.45 kcal/mol)	2/10/2026
IBM Torino (133q)	Hardware	-1.0976 Ha (24.91 kcal/mol)	2/10/2026
QI Tuna-9 (9q)	Hardware	-1.0045 Ha (83.3 kcal/mol)	2/10/2026
IBM Marrakesh (156q)	Hardware	-1.0956 Ha (26.1505 kcal/mol)	2/10/2026
ibm-torino	Hardware	-1.0408 Ha (60.54 kcal/mol)	Invalid Date
QI Tuna-9 (9q)	Hardware	--	Invalid Date

H₂ Dissociation Curve

14 emulator points + 7 hardware points

Energy vs. bond distance for molecular hydrogen. The VQE emulator matches the exact (FCI) curve within chemical accuracy at all 14 distances. Purple diamonds show Tuna-9 hardware results with readout error mitigation (sector-projected 2-qubit ansatz, native CZ gate set). Error bars show ±1σ across 5 independent hardware runs.

Emulator eq. energy: -1.1378 Ha

Emulator max error: 0.84 kcal/mol

Hardware eq. energy: -1.1324 Ha ±11.4 mHa

Hardware mean error: 1.8 kcal/mol

Repetitions: 5

IBM Torino (133q)vqe-equilibrium-001-ibm

completed

Measured Energy

-1.1226 Ha

FCI Reference

-1.1373 Ha

Error:0.0147 Ha (9.22 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.961

<Z1>

0.950

<Z0Z1>

-0.969

<X0X1>

-0.256

<Y0Y1>

-0.197

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|01⟩

3973 (97.0%)

|10⟩

59 (1.4%)

|11⟩

43 (1.0%)

|00⟩

21 (0.5%)

X-basisHadamard rotated4096 shots

|01⟩

1296 (31.6%)

|10⟩

1276 (31.2%)

|00⟩

802 (19.6%)

|11⟩

722 (17.6%)

Y-basisS+H rotated4096 shots

|01⟩

1242 (30.3%)

|10⟩

1209 (29.5%)

|11⟩

832 (20.3%)

|00⟩

813 (19.8%)

VQE energy: -1.1226 Ha (FCI: -1.1373 Ha). Error: 9.22 kcal/mol. Not chemical accuracy — hardware noise degrades Z correlations by ~3-5%.

View raw JSON

IBM Torino (133q)vqe-h2-sagastizabal-ibm

completed

Measured Energy

-1.1197 Ha

FCI Reference

-1.1373 Ha

Error:0.0176 Ha (11 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.956

<Z1>

0.949

<Z0Z1>

-0.971

<X0X1>

-0.218

<Y0Y1>

-0.229

IBM Torino H2 VQE: -1.1197 Ha, 11.0 kcal/mol error. Hardware noise degrades result but correct sector and qualitative behavior observed.

View raw JSON

iqm-garnetvqe-iqm-garnet-rem-001

completed

Measured Energy

-1.1146 Ha

FCI Reference

-1.1373 Ha

Error:0.0227 Ha (14.26 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.949

<Z1>

0.949

<Z0Z1>

-1.000

<X0X1>

-0.206

<Y0Y1>

-0.196

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|10⟩

3912 (95.5%)

|01⟩

102 (2.5%)

|11⟩

42 (1.0%)

|00⟩

40 (1.0%)

X-basisHadamard rotated4096 shots

|01⟩

1270 (31.0%)

|10⟩

1200 (29.3%)

|00⟩

826 (20.2%)

|11⟩

800 (19.5%)

Y-basisS+H rotated4096 shots

|01⟩

1277 (31.2%)

|10⟩

1172 (28.6%)

|11⟩

875 (21.4%)

|00⟩

772 (18.8%)

View raw JSON

QI Tuna-9 (9q)vqe-h2-sagastizabal-tuna9-q68

completed

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|001000000⟩

3612 (88.2%)

|000000000⟩

137 (3.3%)

|001000001⟩

100 (2.4%)

|011000000⟩

68 (1.7%)

|100000000⟩

44 (1.1%)

|001010000⟩

40 (1.0%)

|001100000⟩

26 (0.6%)

|001000100⟩

21 (0.5%)

X-basisHadamard rotated4096 shots

|001000000⟩

1132 (27.6%)

|100000000⟩

1059 (25.9%)

|000000000⟩

828 (20.2%)

|101000000⟩

705 (17.2%)

|110000000⟩

37 (0.9%)

|100000001⟩

33 (0.8%)

|001000001⟩

29 (0.7%)

|010000000⟩

29 (0.7%)

Y-basisS+H rotated4096 shots

|100000000⟩

1180 (28.8%)

|001000000⟩

1165 (28.4%)

|000000000⟩

769 (18.8%)

|101000000⟩

725 (17.7%)

|100000001⟩

28 (0.7%)

|001000001⟩

24 (0.6%)

|101000001⟩

22 (0.5%)

|000000001⟩

19 (0.5%)

View raw JSON

QI Emulatorvqe-equilibrium-001

completed

Measured Energy

-1.1385 Ha

FCI Reference

-1.1373 Ha

Error:0.0012 Ha (0.75 kcal/mol)Chemical Accuracy

Energy Level Diagram

Expectation Values

<Z0>

-0.973

<Z1>

0.973

<Z0Z1>

-1.000

<X0X1>

-0.252

<Y0Y1>

-0.219

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|01⟩

4041 (98.7%)

|10⟩

55 (1.3%)

X-basisHadamard rotated4096 shots

|10⟩

1295 (31.6%)

|01⟩

1270 (31.0%)

|11⟩

777 (19.0%)

|00⟩

754 (18.4%)

Y-basisS+H rotated4096 shots

|10⟩

1278 (31.2%)

|01⟩

1218 (29.7%)

|11⟩

808 (19.7%)

|00⟩

792 (19.3%)

VQE energy: -1.1385 Ha (FCI: -1.1373 Ha). Error: 0.75 kcal/mol. Within chemical accuracy.

View cQASM circuit

version 3.0
qubit[2] q
bit[2] b

// Subspace-preserving ansatz: Ry-CNOT-X
// State = cos(a/2)|10> + sin(a/2)|01>
Ry(-0.223539) q[0]
CNOT q[0], q[1]
X q[0]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-rem-003

completed

Measured Energy

-1.1310 Ha

FCI Reference

-1.1373 Ha

Error:0.0063 Ha (3.94 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.963

<Z1>

0.963

<Z0Z1>

-1.000

<X0X1>

-0.209

<Y0Y1>

-0.200

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3221 (78.6%)

|00000⟩

593 (14.5%)

|10100⟩

68 (1.7%)

|00101⟩

65 (1.6%)

|10000⟩

61 (1.5%)

|00110⟩

39 (1.0%)

|01100⟩

23 (0.6%)

|00010⟩

12 (0.3%)

X-basisHadamard rotated4096 shots

|10000⟩

1199 (29.3%)

|00100⟩

1181 (28.8%)

|00000⟩

902 (22.0%)

|10100⟩

641 (15.6%)

|10001⟩

28 (0.7%)

|00110⟩

19 (0.5%)

|00101⟩

18 (0.4%)

|00001⟩

17 (0.4%)

Y-basisS+H rotated4096 shots

|00100⟩

1259 (30.7%)

|10000⟩

1077 (26.3%)

|00000⟩

852 (20.8%)

|10100⟩

695 (17.0%)

|10010⟩

34 (0.8%)

|00010⟩

30 (0.7%)

|00110⟩

27 (0.7%)

|00101⟩

25 (0.6%)

VQE energy: -1.1310 Ha (FCI: -1.1373 Ha). Error: 3.9 kcal/mol. Outside chemical accuracy. Post-selection kept 83% of Z-basis shots. Hybrid(PS+REM): -1.1310 Ha (3.9 kcal/mol). Full-REM: -1.0751 Ha (39.0 kcal/mol).

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X (cnot_folds=1)
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223400
Ry(-0.223400) q[2]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-rem-002

completed

Measured Energy

-1.1293 Ha

FCI Reference

-1.1373 Ha

Error:0.0080 Ha (5.01 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.964

<Z1>

0.964

<Z0Z1>

-1.000

<X0X1>

-0.202

<Y0Y1>

-0.186

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3535 (86.3%)

|00000⟩

271 (6.6%)

|10100⟩

80 (2.0%)

|00101⟩

75 (1.8%)

|10000⟩

66 (1.6%)

|00110⟩

43 (1.0%)

|01100⟩

16 (0.4%)

|00001⟩

3 (0.1%)

X-basisHadamard rotated4096 shots

|10000⟩

1179 (28.8%)

|00100⟩

1175 (28.7%)

|00000⟩

851 (20.8%)

|10100⟩

718 (17.5%)

|10001⟩

25 (0.6%)

|00101⟩

24 (0.6%)

|10010⟩

22 (0.5%)

|00001⟩

21 (0.5%)

Y-basisS+H rotated4096 shots

|00100⟩

1219 (29.8%)

|10000⟩

1104 (27.0%)

|00000⟩

874 (21.3%)

|10100⟩

717 (17.5%)

|00110⟩

33 (0.8%)

|10010⟩

23 (0.6%)

|00101⟩

23 (0.6%)

|10110⟩

18 (0.4%)

VQE energy: -1.1293 Ha (FCI: -1.1373 Ha). Error: 5.0 kcal/mol. Outside chemical accuracy. Post-selection kept 91% of Z-basis shots. Hybrid(PS+REM): -1.1293 Ha (5.0 kcal/mol). Full-REM: -1.1277 Ha (6.0 kcal/mol).

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X (cnot_folds=1)
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223400
Ry(-0.223400) q[2]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-rem-001

completed

Measured Energy

-1.1358 Ha

FCI Reference

-1.1373 Ha

Error:0.0015 Ha (0.92 kcal/mol)Chemical Accuracy

Energy Level Diagram

Expectation Values

<Z0>

-0.969

<Z1>

0.969

<Z0Z1>

-1.000

<X0X1>

-0.210

<Y0Y1>

-0.207

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3737 (91.2%)

|10100⟩

85 (2.1%)

|00000⟩

71 (1.7%)

|00101⟩

65 (1.6%)

|10000⟩

60 (1.5%)

|00110⟩

52 (1.3%)

|01100⟩

16 (0.4%)

|10101⟩

3 (0.1%)

X-basisHadamard rotated4096 shots

|00100⟩

1227 (30.0%)

|10000⟩

1171 (28.6%)

|00000⟩

823 (20.1%)

|10100⟩

738 (18.0%)

|00101⟩

26 (0.6%)

|10001⟩

17 (0.4%)

|10010⟩

16 (0.4%)

|10110⟩

14 (0.3%)

Y-basisS+H rotated4096 shots

|10000⟩

1235 (30.2%)

|00100⟩

1131 (27.6%)

|00000⟩

861 (21.0%)

|10100⟩

691 (16.9%)

|10010⟩

31 (0.8%)

|00110⟩

26 (0.6%)

|10001⟩

23 (0.6%)

|00010⟩

21 (0.5%)

VQE energy: -1.1358 Ha (FCI: -1.1373 Ha). Error: 0.9 kcal/mol. Within chemical accuracy. Post-selection kept 96% of Z-basis shots. Hybrid(PS+REM): -1.1358 Ha (0.9 kcal/mol). Full-REM: -1.1325 Ha (3.0 kcal/mol).

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X (cnot_folds=1)
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223400
Ry(-0.223400) q[2]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-zne-f5-004

completed

Measured Energy

-1.1213 Ha

FCI Reference

-1.1373 Ha

Error:0.0160 Ha (10.02 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.960

<Z1>

0.960

<Z0Z1>

-1.000

<X0X1>

-0.217

<Y0Y1>

-0.185

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3617 (88.3%)

|10100⟩

148 (3.6%)

|00000⟩

112 (2.7%)

|10000⟩

77 (1.9%)

|00101⟩

72 (1.8%)

|00110⟩

40 (1.0%)

|00001⟩

9 (0.2%)

|01100⟩

9 (0.2%)

X-basisHadamard rotated4096 shots

|00100⟩

1220 (29.8%)

|10000⟩

1176 (28.7%)

|00000⟩

801 (19.6%)

|10100⟩

743 (18.1%)

|00101⟩

26 (0.6%)

|10010⟩

23 (0.6%)

|10001⟩

18 (0.4%)

|00110⟩

18 (0.4%)

Y-basisS+H rotated4096 shots

|00100⟩

1196 (29.2%)

|10000⟩

1127 (27.5%)

|00000⟩

799 (19.5%)

|10100⟩

794 (19.4%)

|10010⟩

29 (0.7%)

|00110⟩

27 (0.7%)

|10110⟩

22 (0.5%)

|00010⟩

20 (0.5%)

VQE energy: -1.1213 Ha (FCI: -1.1373 Ha). Error: 10.0 kcal/mol. Outside chemical accuracy. Post-selection kept 93% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X (cnot_folds=5)
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223400
Ry(-0.223400) q[2]
CNOT q[2], q[4]
CNOT q[2], q[4]
CNOT q[2], q[4]
CNOT q[2], q[4]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-zne-f5-003

completed

Measured Energy

-1.1329 Ha

FCI Reference

-1.1373 Ha

Error:0.0044 Ha (2.79 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.976

<Z1>

0.976

<Z0Z1>

-1.000

<X0X1>

-0.198

<Y0Y1>

-0.186

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3690 (90.1%)

|10100⟩

147 (3.6%)

|00000⟩

75 (1.8%)

|00101⟩

66 (1.6%)

|00110⟩

56 (1.4%)

|10000⟩

45 (1.1%)

|01100⟩

6 (0.1%)

|10101⟩

3 (0.1%)

X-basisHadamard rotated4096 shots

|10000⟩

1232 (30.1%)

|00100⟩

1153 (28.1%)

|00000⟩

864 (21.1%)

|10100⟩

727 (17.7%)

|00101⟩

18 (0.4%)

|10001⟩

17 (0.4%)

|00010⟩

17 (0.4%)

|10010⟩

15 (0.4%)

Y-basisS+H rotated4096 shots

|10000⟩

1230 (30.0%)

|00100⟩

1095 (26.7%)

|10100⟩

812 (19.8%)

|00000⟩

779 (19.0%)

|00110⟩

29 (0.7%)

|00101⟩

28 (0.7%)

|10110⟩

24 (0.6%)

|00010⟩

22 (0.5%)

VQE energy: -1.1329 Ha (FCI: -1.1373 Ha). Error: 2.8 kcal/mol. Outside chemical accuracy. Post-selection kept 94% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X (cnot_folds=5)
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223400
Ry(-0.223400) q[2]
CNOT q[2], q[4]
CNOT q[2], q[4]
CNOT q[2], q[4]
CNOT q[2], q[4]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-zne-f5-002

completed

Measured Energy

-1.1275 Ha

FCI Reference

-1.1373 Ha

Error:0.0098 Ha (6.14 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.972

<Z1>

0.972

<Z0Z1>

-1.000

<X0X1>

-0.193

<Y0Y1>

-0.171

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3731 (91.1%)

|10100⟩

118 (2.9%)

|00101⟩

74 (1.8%)

|00000⟩

61 (1.5%)

|10000⟩

55 (1.3%)

|00110⟩

44 (1.1%)

|01100⟩

6 (0.1%)

|10101⟩

2 (0.0%)

X-basisHadamard rotated4096 shots

|00100⟩

1188 (29.0%)

|10000⟩

1174 (28.7%)

|00000⟩

835 (20.4%)

|10100⟩

751 (18.3%)

|00101⟩

29 (0.7%)

|00110⟩

20 (0.5%)

|00001⟩

19 (0.5%)

|10001⟩

17 (0.4%)

Y-basisS+H rotated4096 shots

|00100⟩

1165 (28.4%)

|10000⟩

1132 (27.6%)

|00000⟩

865 (21.1%)

|10100⟩

763 (18.6%)

|10010⟩

32 (0.8%)

|00101⟩

25 (0.6%)

|00010⟩

21 (0.5%)

|00110⟩

21 (0.5%)

VQE energy: -1.1275 Ha (FCI: -1.1373 Ha). Error: 6.1 kcal/mol. Outside chemical accuracy. Post-selection kept 95% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X (cnot_folds=5)
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223400
Ry(-0.223400) q[2]
CNOT q[2], q[4]
CNOT q[2], q[4]
CNOT q[2], q[4]
CNOT q[2], q[4]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-zne-f3-004

completed

Measured Energy

-1.1263 Ha

FCI Reference

-1.1373 Ha

Error:0.0110 Ha (6.93 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.968

<Z1>

0.968

<Z0Z1>

-1.000

<X0X1>

-0.183

<Y0Y1>

-0.204

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3695 (90.2%)

|00000⟩

96 (2.3%)

|10100⟩

94 (2.3%)

|00101⟩

79 (1.9%)

|10000⟩

62 (1.5%)

|00110⟩

51 (1.2%)

|01100⟩

9 (0.2%)

|10101⟩

3 (0.1%)

X-basisHadamard rotated4096 shots

|10000⟩

1209 (29.5%)

|00100⟩

1131 (27.6%)

|00000⟩

922 (22.5%)

|10100⟩

686 (16.7%)

|00001⟩

21 (0.5%)

|10001⟩

20 (0.5%)

|00101⟩

20 (0.5%)

|00110⟩

18 (0.4%)

Y-basisS+H rotated4096 shots

|00100⟩

1205 (29.4%)

|10000⟩

1166 (28.5%)

|00000⟩

797 (19.5%)

|10100⟩

769 (18.8%)

|00101⟩

27 (0.7%)

|10010⟩

22 (0.5%)

|10001⟩

20 (0.5%)

|00110⟩

20 (0.5%)

VQE energy: -1.1263 Ha (FCI: -1.1373 Ha). Error: 6.9 kcal/mol. Outside chemical accuracy. Post-selection kept 95% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X (cnot_folds=3)
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223400
Ry(-0.223400) q[2]
CNOT q[2], q[4]
CNOT q[2], q[4]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-zne-f3-003

completed

Measured Energy

-1.1160 Ha

FCI Reference

-1.1373 Ha

Error:0.0214 Ha (13.4 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.964

<Z1>

0.964

<Z0Z1>

-1.000

<X0X1>

-0.196

<Y0Y1>

-0.110

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3714 (90.7%)

|10100⟩

110 (2.7%)

|00101⟩

75 (1.8%)

|10000⟩

71 (1.7%)

|00000⟩

52 (1.3%)

|00110⟩

50 (1.2%)

|01100⟩

13 (0.3%)

|00001⟩

4 (0.1%)

X-basisHadamard rotated4096 shots

|00100⟩

1230 (30.0%)

|10000⟩

1145 (28.0%)

|00000⟩

858 (20.9%)

|10100⟩

732 (17.9%)

|00101⟩

18 (0.4%)

|10010⟩

17 (0.4%)

|10101⟩

16 (0.4%)

|00001⟩

15 (0.4%)

Y-basisS+H rotated4096 shots

|10000⟩

1327 (32.4%)

|00000⟩

1191 (29.1%)

|00100⟩

851 (20.8%)

|10100⟩

541 (13.2%)

|00010⟩

33 (0.8%)

|10010⟩

31 (0.8%)

|10001⟩

29 (0.7%)

|00001⟩

29 (0.7%)

VQE energy: -1.1160 Ha (FCI: -1.1373 Ha). Error: 13.4 kcal/mol. Outside chemical accuracy. Post-selection kept 96% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X (cnot_folds=3)
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223400
Ry(-0.223400) q[2]
CNOT q[2], q[4]
CNOT q[2], q[4]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-zne-f3-002

completed

Measured Energy

-1.1291 Ha

FCI Reference

-1.1373 Ha

Error:0.0082 Ha (5.13 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.967

<Z1>

0.967

<Z0Z1>

-1.000

<X0X1>

-0.231

<Y0Y1>

-0.190

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3729 (91.0%)

|10100⟩

111 (2.7%)

|00101⟩

70 (1.7%)

|10000⟩

64 (1.6%)

|00000⟩

60 (1.5%)

|00110⟩

43 (1.0%)

|01100⟩

13 (0.3%)

|00001⟩

3 (0.1%)

X-basisHadamard rotated4096 shots

|00100⟩

1231 (30.1%)

|10000⟩

1195 (29.2%)

|00000⟩

826 (20.2%)

|10100⟩

699 (17.1%)

|00101⟩

30 (0.7%)

|10001⟩

25 (0.6%)

|00110⟩

22 (0.5%)

|10101⟩

15 (0.4%)

Y-basisS+H rotated4096 shots

|00100⟩

1250 (30.5%)

|10000⟩

1071 (26.1%)

|00000⟩

876 (21.4%)

|10100⟩

717 (17.5%)

|10010⟩

30 (0.7%)

|00110⟩

30 (0.7%)

|00101⟩

26 (0.6%)

|10001⟩

21 (0.5%)

VQE energy: -1.1291 Ha (FCI: -1.1373 Ha). Error: 5.1 kcal/mol. Outside chemical accuracy. Post-selection kept 96% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X (cnot_folds=3)
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223400
Ry(-0.223400) q[2]
CNOT q[2], q[4]
CNOT q[2], q[4]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-zne-f1-004

completed

Measured Energy

-1.1294 Ha

FCI Reference

-1.1373 Ha

Error:0.0079 Ha (4.94 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.967

<Z1>

0.967

<Z0Z1>

-1.000

<X0X1>

-0.217

<Y0Y1>

-0.211

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3671 (89.6%)

|00000⟩

125 (3.1%)

|10100⟩

85 (2.1%)

|00101⟩

82 (2.0%)

|10000⟩

64 (1.6%)

|00110⟩

42 (1.0%)

|01100⟩

11 (0.3%)

|00001⟩

7 (0.2%)

X-basisHadamard rotated4096 shots

|00100⟩

1205 (29.4%)

|10000⟩

1204 (29.4%)

|00000⟩

800 (19.5%)

|10100⟩

745 (18.2%)

|10001⟩

23 (0.6%)

|00001⟩

16 (0.4%)

|00101⟩

16 (0.4%)

|10010⟩

15 (0.4%)

Y-basisS+H rotated4096 shots

|10000⟩

1200 (29.3%)

|00100⟩

1172 (28.6%)

|00000⟩

870 (21.2%)

|10100⟩

681 (16.6%)

|00101⟩

26 (0.6%)

|00110⟩

24 (0.6%)

|10001⟩

23 (0.6%)

|10010⟩

21 (0.5%)

VQE energy: -1.1294 Ha (FCI: -1.1373 Ha). Error: 4.9 kcal/mol. Outside chemical accuracy. Post-selection kept 95% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X (cnot_folds=1)
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223400
Ry(-0.223400) q[2]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-zne-f1-003

completed

Measured Energy

-1.1213 Ha

FCI Reference

-1.1373 Ha

Error:0.0160 Ha (10.02 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.963

<Z1>

0.963

<Z0Z1>

-1.000

<X0X1>

-0.149

<Y0Y1>

-0.223

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3693 (90.2%)

|00000⟩

104 (2.5%)

|00101⟩

82 (2.0%)

|10100⟩

73 (1.8%)

|10000⟩

69 (1.7%)

|00110⟩

51 (1.2%)

|01100⟩

16 (0.4%)

|10010⟩

2 (0.0%)

X-basisHadamard rotated4096 shots

|10000⟩

1199 (29.3%)

|00100⟩

1051 (25.7%)

|00000⟩

987 (24.1%)

|10100⟩

685 (16.7%)

|00101⟩

26 (0.6%)

|10001⟩

24 (0.6%)

|00110⟩

20 (0.5%)

|00001⟩

16 (0.4%)

Y-basisS+H rotated4096 shots

|00100⟩

1206 (29.4%)

|10000⟩

1182 (28.9%)

|00000⟩

830 (20.3%)

|10100⟩

692 (16.9%)

|00110⟩

33 (0.8%)

|10010⟩

28 (0.7%)

|00101⟩

25 (0.6%)

|10001⟩

22 (0.5%)

VQE energy: -1.1213 Ha (FCI: -1.1373 Ha). Error: 10.0 kcal/mol. Outside chemical accuracy. Post-selection kept 96% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X (cnot_folds=1)
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223400
Ry(-0.223400) q[2]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-zne-f1-002

completed

Measured Energy

-1.1221 Ha

FCI Reference

-1.1373 Ha

Error:0.0152 Ha (9.53 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.967

<Z1>

0.967

<Z0Z1>

-1.000

<X0X1>

-0.191

<Y0Y1>

-0.154

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3721 (90.8%)

|00000⟩

105 (2.6%)

|10100⟩

92 (2.2%)

|10000⟩

62 (1.5%)

|00101⟩

59 (1.4%)

|00110⟩

39 (1.0%)

|01100⟩

12 (0.3%)

|10101⟩

3 (0.1%)

X-basisHadamard rotated4096 shots

|00100⟩

1213 (29.6%)

|10000⟩

1137 (27.8%)

|00000⟩

829 (20.2%)

|10100⟩

758 (18.5%)

|10001⟩

25 (0.6%)

|00101⟩

23 (0.6%)

|10101⟩

20 (0.5%)

|00001⟩

16 (0.4%)

Y-basisS+H rotated4096 shots

|00100⟩

1180 (28.8%)

|10000⟩

1078 (26.3%)

|00000⟩

889 (21.7%)

|10100⟩

750 (18.3%)

|10010⟩

31 (0.8%)

|00001⟩

28 (0.7%)

|00110⟩

27 (0.7%)

|00010⟩

23 (0.6%)

VQE energy: -1.1221 Ha (FCI: -1.1373 Ha). Error: 9.5 kcal/mol. Outside chemical accuracy. Post-selection kept 95% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X (cnot_folds=1)
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223400
Ry(-0.223400) q[2]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-zne-f5-001

completed

Measured Energy

-1.1238 Ha

FCI Reference

-1.1373 Ha

Error:0.0135 Ha (8.5 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.968

<Z1>

0.968

<Z0Z1>

-1.000

<X0X1>

-0.185

<Y0Y1>

-0.172

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3626 (88.5%)

|10100⟩

170 (4.2%)

|00000⟩

109 (2.7%)

|00101⟩

72 (1.8%)

|10000⟩

61 (1.5%)

|00110⟩

33 (0.8%)

|01100⟩

8 (0.2%)

|10110⟩

5 (0.1%)

X-basisHadamard rotated4096 shots

|00100⟩

1182 (28.9%)

|10000⟩

1152 (28.1%)

|00000⟩

887 (21.7%)

|10100⟩

732 (17.9%)

|00101⟩

29 (0.7%)

|10001⟩

24 (0.6%)

|00001⟩

17 (0.4%)

|00110⟩

17 (0.4%)

Y-basisS+H rotated4096 shots

|00100⟩

1249 (30.5%)

|10000⟩

1050 (25.6%)

|00000⟩

883 (21.6%)

|10100⟩

738 (18.0%)

|10010⟩

29 (0.7%)

|00110⟩

25 (0.6%)

|00101⟩

22 (0.5%)

|00010⟩

21 (0.5%)

VQE energy: -1.1238 Ha (FCI: -1.1373 Ha). Error: 8.5 kcal/mol. Outside chemical accuracy. Post-selection kept 93% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X (cnot_folds=5)
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223400
Ry(-0.223400) q[2]
CNOT q[2], q[4]
CNOT q[2], q[4]
CNOT q[2], q[4]
CNOT q[2], q[4]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-zne-f3-001

completed

Measured Energy

-1.1230 Ha

FCI Reference

-1.1373 Ha

Error:0.0143 Ha (9 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.964

<Z1>

0.964

<Z0Z1>

-1.000

<X0X1>

-0.182

<Y0Y1>

-0.202

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3676 (89.7%)

|00000⟩

105 (2.6%)

|10100⟩

103 (2.5%)

|00101⟩

77 (1.9%)

|10000⟩

66 (1.6%)

|00110⟩

36 (0.9%)

|01100⟩

12 (0.3%)

|00010⟩

6 (0.1%)

X-basisHadamard rotated4096 shots

|00100⟩

1188 (29.0%)

|10000⟩

1141 (27.9%)

|00000⟩

843 (20.6%)

|10100⟩

773 (18.9%)

|00101⟩

29 (0.7%)

|00110⟩

27 (0.7%)

|10101⟩

21 (0.5%)

|00001⟩

19 (0.5%)

Y-basisS+H rotated4096 shots

|00100⟩

1271 (31.0%)

|10000⟩

1089 (26.6%)

|00000⟩

862 (21.0%)

|10100⟩

705 (17.2%)

|00101⟩

27 (0.7%)

|00110⟩

27 (0.7%)

|10001⟩

21 (0.5%)

|00010⟩

21 (0.5%)

VQE energy: -1.1230 Ha (FCI: -1.1373 Ha). Error: 9.0 kcal/mol. Outside chemical accuracy. Post-selection kept 95% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X (cnot_folds=3)
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223400
Ry(-0.223400) q[2]
CNOT q[2], q[4]
CNOT q[2], q[4]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-zne-f1-001

completed

Measured Energy

-1.1272 Ha

FCI Reference

-1.1373 Ha

Error:0.0101 Ha (6.31 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.967

<Z1>

0.967

<Z0Z1>

-1.000

<X0X1>

-0.215

<Y0Y1>

-0.189

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3678 (89.8%)

|00000⟩

103 (2.5%)

|10100⟩

98 (2.4%)

|00101⟩

90 (2.2%)

|10000⟩

62 (1.5%)

|00110⟩

38 (0.9%)

|01100⟩

11 (0.3%)

|10101⟩

5 (0.1%)

X-basisHadamard rotated4096 shots

|00100⟩

1197 (29.2%)

|10000⟩

1186 (29.0%)

|00000⟩

781 (19.1%)

|10100⟩

767 (18.7%)

|10001⟩

30 (0.7%)

|10010⟩

20 (0.5%)

|00101⟩

19 (0.5%)

|00110⟩

19 (0.5%)

Y-basisS+H rotated4096 shots

|00100⟩

1211 (29.6%)

|10000⟩

1123 (27.4%)

|00000⟩

894 (21.8%)

|10100⟩

705 (17.2%)

|00110⟩

31 (0.8%)

|10010⟩

24 (0.6%)

|00101⟩

20 (0.5%)

|10001⟩

18 (0.4%)

VQE energy: -1.1272 Ha (FCI: -1.1373 Ha). Error: 6.3 kcal/mol. Outside chemical accuracy. Post-selection kept 95% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X (cnot_folds=1)
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223400
Ry(-0.223400) q[2]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Emulatorvqe-emulator-zne-f5-001

completed

Measured Energy

-1.1296 Ha

FCI Reference

-1.1373 Ha

Error:0.0077 Ha (4.83 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.968

<Z1>

0.968

<Z0Z1>

-1.000

<X0X1>

-0.203

<Y0Y1>

-0.220

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|01⟩

4030 (98.4%)

|10⟩

66 (1.6%)

X-basisHadamard rotated4096 shots

|10⟩

1238 (30.2%)

|01⟩

1226 (29.9%)

|00⟩

850 (20.8%)

|11⟩

782 (19.1%)

Y-basisS+H rotated4096 shots

|01⟩

1268 (31.0%)

|10⟩

1230 (30.0%)

|00⟩

885 (21.6%)

|11⟩

713 (17.4%)

VQE energy: -1.1296 Ha (FCI: -1.1373 Ha). Error: 4.8 kcal/mol. Outside chemical accuracy. Post-selection kept 100% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[2] q
bit[2] b

// Subspace-preserving ansatz: Ry-CNOT-X (cnot_folds=5)
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223400
Ry(-0.223400) q[0]
CNOT q[0], q[1]
CNOT q[0], q[1]
CNOT q[0], q[1]
CNOT q[0], q[1]
CNOT q[0], q[1]
X q[0]

// Z-basis measurement
b = measure q

View raw JSON

QI Emulatorvqe-emulator-zne-f3-001

completed

Measured Energy

-1.1415 Ha

FCI Reference

-1.1373 Ha

Error:0.0042 Ha (2.64 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.978

<Z1>

0.978

<Z0Z1>

-1.000

<X0X1>

-0.246

<Y0Y1>

-0.219

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|01⟩

4051 (98.9%)

|10⟩

45 (1.1%)

X-basisHadamard rotated4096 shots

|01⟩

1308 (31.9%)

|10⟩

1243 (30.3%)

|00⟩

775 (18.9%)

|11⟩

770 (18.8%)

Y-basisS+H rotated4096 shots

|01⟩

1252 (30.6%)

|10⟩

1244 (30.4%)

|00⟩

813 (19.8%)

|11⟩

787 (19.2%)

VQE energy: -1.1415 Ha (FCI: -1.1373 Ha). Error: 2.6 kcal/mol. Outside chemical accuracy. Post-selection kept 100% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[2] q
bit[2] b

// Subspace-preserving ansatz: Ry-CNOT-X (cnot_folds=3)
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223400
Ry(-0.223400) q[0]
CNOT q[0], q[1]
CNOT q[0], q[1]
CNOT q[0], q[1]
X q[0]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-sweep-250

completed

Measured Energy

-0.9147 Ha

FCI Reference

-0.9361 Ha

Error:0.0214 Ha (13.42 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.205

<Z1>

0.205

<Z0Z1>

-1.000

<X0X1>

-0.914

<Y0Y1>

-0.892

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

2264 (55.3%)

|10000⟩

1481 (36.2%)

|00000⟩

83 (2.0%)

|10100⟩

78 (1.9%)

|00101⟩

48 (1.2%)

|00110⟩

39 (1.0%)

|10010⟩

37 (0.9%)

|10001⟩

34 (0.8%)

X-basisHadamard rotated4096 shots

|00100⟩

1924 (47.0%)

|10000⟩

1818 (44.4%)

|00000⟩

102 (2.5%)

|10100⟩

67 (1.6%)

|10001⟩

43 (1.0%)

|10010⟩

37 (0.9%)

|00110⟩

37 (0.9%)

|00101⟩

36 (0.9%)

Y-basisS+H rotated4096 shots

|10000⟩

1882 (45.9%)

|00100⟩

1785 (43.6%)

|00000⟩

147 (3.6%)

|00110⟩

68 (1.7%)

|10010⟩

64 (1.6%)

|10100⟩

62 (1.5%)

|10001⟩

34 (0.8%)

|00101⟩

30 (0.7%)

VQE energy: -0.9147 Ha (FCI: -0.9361 Ha). Error: 13.4 kcal/mol. Outside chemical accuracy. Post-selection kept 96% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-1.380900
Ry(-1.380900) q[2]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-bestpair-001

completed

Measured Energy

-1.1325 Ha

FCI Reference

-1.1373 Ha

Error:0.0048 Ha (3.04 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.974

<Z1>

0.974

<Z0Z1>

-1.000

<X0X1>

-0.180

<Y0Y1>

-0.219

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3756 (91.7%)

|00000⟩

86 (2.1%)

|00101⟩

73 (1.8%)

|10100⟩

67 (1.6%)

|10000⟩

50 (1.2%)

|00110⟩

37 (0.9%)

|01100⟩

17 (0.4%)

|00001⟩

5 (0.1%)

X-basisHadamard rotated4096 shots

|10000⟩

1169 (28.5%)

|00100⟩

1163 (28.4%)

|00000⟩

846 (20.7%)

|10100⟩

789 (19.3%)

|00101⟩

25 (0.6%)

|10001⟩

20 (0.5%)

|10101⟩

13 (0.3%)

|00001⟩

12 (0.3%)

Y-basisS+H rotated4096 shots

|00100⟩

1203 (29.4%)

|10000⟩

1167 (28.5%)

|00000⟩

812 (19.8%)

|10100⟩

721 (17.6%)

|10001⟩

31 (0.8%)

|10010⟩

30 (0.7%)

|00110⟩

28 (0.7%)

|00101⟩

27 (0.7%)

VQE energy: -1.1325 Ha (FCI: -1.1373 Ha). Error: 3.0 kcal/mol. Outside chemical accuracy. Post-selection kept 96% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223400
Ry(-0.223400) q[2]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Emulatorvqe-emulator-sweep-250

completed

Measured Energy

-0.9362 Ha

FCI Reference

-0.9361 Ha

Error:0.0001 Ha (0.09 kcal/mol)Chemical Accuracy

Energy Level Diagram

Expectation Values

<Z0>

-0.183

<Z1>

0.183

<Z0Z1>

-1.000

<X0X1>

-0.983

<Y0Y1>

-0.984

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|01⟩

2423 (59.2%)

|10⟩

1673 (40.8%)

X-basisHadamard rotated4096 shots

|10⟩

2032 (49.6%)

|01⟩

2029 (49.5%)

|11⟩

20 (0.5%)

|00⟩

15 (0.4%)

Y-basisS+H rotated4096 shots

|01⟩

2087 (51.0%)

|10⟩

1977 (48.3%)

|11⟩

20 (0.5%)

|00⟩

12 (0.3%)

View raw JSON

QI Emulatorvqe-emulator-sweep-200

completed

Measured Energy

-0.9454 Ha

FCI Reference

-0.9486 Ha

Error:0.0033 Ha (2.05 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.399

<Z1>

0.399

<Z0Z1>

-1.000

<X0X1>

-0.915

<Y0Y1>

-0.895

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|01⟩

2866 (70.0%)

|10⟩

1230 (30.0%)

X-basisHadamard rotated4096 shots

|10⟩

1971 (48.1%)

|01⟩

1950 (47.6%)

|00⟩

90 (2.2%)

|11⟩

85 (2.1%)

Y-basisS+H rotated4096 shots

|10⟩

1986 (48.5%)

|01⟩

1894 (46.2%)

|11⟩

115 (2.8%)

|00⟩

101 (2.5%)

View raw JSON

QI Emulatorvqe-emulator-sweep-150

completed

Measured Energy

-0.9940 Ha

FCI Reference

-0.9981 Ha

Error:0.0042 Ha (2.62 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.742

<Z1>

0.742

<Z0Z1>

-1.000

<X0X1>

-0.646

<Y0Y1>

-0.659

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|01⟩

3567 (87.1%)

|10⟩

529 (12.9%)

X-basisHadamard rotated4096 shots

|10⟩

1753 (42.8%)

|01⟩

1619 (39.5%)

|11⟩

385 (9.4%)

|00⟩

339 (8.3%)

Y-basisS+H rotated4096 shots

|10⟩

1725 (42.1%)

|01⟩

1672 (40.8%)

|11⟩

363 (8.9%)

|00⟩

336 (8.2%)

View raw JSON

QI Emulatorvqe-emulator-sweep-100

completed

Measured Energy

-1.1034 Ha

FCI Reference

-1.1012 Ha

Error:0.0022 Ha (1.4 kcal/mol)Chemical Accuracy

Energy Level Diagram

Expectation Values

<Z0>

-0.939

<Z1>

0.939

<Z0Z1>

-1.000

<X0X1>

-0.347

<Y0Y1>

-0.361

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|01⟩

3972 (97.0%)

|10⟩

124 (3.0%)

X-basisHadamard rotated4096 shots

|01⟩

1386 (33.8%)

|10⟩

1373 (33.5%)

|11⟩

672 (16.4%)

|00⟩

665 (16.2%)

Y-basisS+H rotated4096 shots

|10⟩

1426 (34.8%)

|01⟩

1361 (33.2%)

|11⟩

664 (16.2%)

|00⟩

645 (15.7%)

View raw JSON

QI Emulatorvqe-emulator-sweep-073

completed

Measured Energy

-1.1364 Ha

FCI Reference

-1.1373 Ha

Error:0.0010 Ha (0.6 kcal/mol)Chemical Accuracy

Energy Level Diagram

Expectation Values

<Z0>

-0.976

<Z1>

0.976

<Z0Z1>

-1.000

<X0X1>

-0.200

<Y0Y1>

-0.229

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|01⟩

4046 (98.8%)

|10⟩

50 (1.2%)

X-basisHadamard rotated4096 shots

|10⟩

1249 (30.5%)

|01⟩

1208 (29.5%)

|00⟩

828 (20.2%)

|11⟩

811 (19.8%)

Y-basisS+H rotated4096 shots

|01⟩

1261 (30.8%)

|10⟩

1256 (30.7%)

|00⟩

790 (19.3%)

|11⟩

789 (19.3%)

View raw JSON

QI Emulatorvqe-emulator-sweep-050

completed

Measured Energy

-1.0583 Ha

FCI Reference

-1.0552 Ha

Error:0.0032 Ha (2 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.992

<Z1>

0.992

<Z0Z1>

-1.000

<X0X1>

-0.149

<Y0Y1>

-0.147

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|01⟩

4079 (99.6%)

|10⟩

17 (0.4%)

X-basisHadamard rotated4096 shots

|10⟩

1177 (28.7%)

|01⟩

1176 (28.7%)

|00⟩

893 (21.8%)

|11⟩

850 (20.8%)

Y-basisS+H rotated4096 shots

|10⟩

1191 (29.1%)

|01⟩

1159 (28.3%)

|11⟩

894 (21.8%)

|00⟩

852 (20.8%)

View raw JSON

iqm-garnetvqe-equilibrium-001-iqm-garnet

completed

Measured Energy

-0.3105 Ha

FCI Reference

-1.1373 Ha

Error:0.8268 Ha (518.85 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.928

<Z1>

-0.939

<Z0Z1>

0.934

<X0X1>

0.174

<Y0Y1>

-0.107

Measurement Counts by Basis

Z-basisComputational basis1024 shots

|11⟩

973 (95.0%)

|10⟩

20 (2.0%)

|00⟩

17 (1.7%)

|01⟩

14 (1.4%)

X-basisHadamard rotated1024 shots

|11⟩

310 (30.3%)

|00⟩

291 (28.4%)

|01⟩

222 (21.7%)

|10⟩

201 (19.6%)

Y-basisS+H rotated1024 shots

|01⟩

305 (29.8%)

|10⟩

262 (25.6%)

|11⟩

242 (23.6%)

|00⟩

215 (21.0%)

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-sweep-200

completed

Measured Energy

-0.9210 Ha

FCI Reference

-0.9486 Ha

Error:0.0276 Ha (17.32 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.437

<Z1>

0.437

<Z0Z1>

-1.000

<X0X1>

-0.759

<Y0Y1>

-0.827

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

2704 (66.0%)

|10000⟩

1055 (25.8%)

|00000⟩

109 (2.7%)

|00110⟩

58 (1.4%)

|10100⟩

54 (1.3%)

|00101⟩

47 (1.1%)

|10010⟩

21 (0.5%)

|10001⟩

20 (0.5%)

X-basisHadamard rotated4096 shots

|00100⟩

1745 (42.6%)

|10000⟩

1688 (41.2%)

|00000⟩

333 (8.1%)

|10100⟩

143 (3.5%)

|10001⟩

44 (1.1%)

|00101⟩

39 (1.0%)

|00110⟩

36 (0.9%)

|10010⟩

25 (0.6%)

Y-basisS+H rotated4096 shots

|10000⟩

1817 (44.4%)

|00100⟩

1722 (42.0%)

|00000⟩

221 (5.4%)

|10100⟩

115 (2.8%)

|00110⟩

64 (1.6%)

|10010⟩

54 (1.3%)

|10001⟩

36 (0.9%)

|00101⟩

26 (0.6%)

VQE energy: -0.9210 Ha (FCI: -0.9486 Ha). Error: 17.3 kcal/mol. Outside chemical accuracy. Post-selection kept 96% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-1.133300
Ry(-1.133300) q[2]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-sweep-150

completed

Measured Energy

-0.9779 Ha

FCI Reference

-0.9981 Ha

Error:0.0202 Ha (12.68 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.711

<Z1>

0.711

<Z0Z1>

-1.000

<X0X1>

-0.624

<Y0Y1>

-0.611

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3204 (78.2%)

|10000⟩

550 (13.4%)

|00101⟩

84 (2.1%)

|10100⟩

78 (1.9%)

|00000⟩

72 (1.8%)

|00110⟩

53 (1.3%)

|01100⟩

26 (0.6%)

|10001⟩

13 (0.3%)

X-basisHadamard rotated4096 shots

|10000⟩

1625 (39.7%)

|00100⟩

1581 (38.6%)

|00000⟩

384 (9.4%)

|10100⟩

351 (8.6%)

|00101⟩

37 (0.9%)

|10001⟩

30 (0.7%)

|00110⟩

20 (0.5%)

|00001⟩

11 (0.3%)

Y-basisS+H rotated4096 shots

|10000⟩

1579 (38.5%)

|00100⟩

1561 (38.1%)

|00000⟩

462 (11.3%)

|10100⟩

297 (7.3%)

|00110⟩

43 (1.0%)

|10010⟩

41 (1.0%)

|10001⟩

32 (0.8%)

|00101⟩

28 (0.7%)

VQE energy: -0.9779 Ha (FCI: -0.9981 Ha). Error: 12.7 kcal/mol. Outside chemical accuracy. Post-selection kept 96% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.726700
Ry(-0.726700) q[2]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-sweep-100

completed

Measured Energy

-1.0946 Ha

FCI Reference

-1.1012 Ha

Error:0.0066 Ha (4.12 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.937

<Z1>

0.937

<Z0Z1>

-1.000

<X0X1>

-0.302

<Y0Y1>

-0.332

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3701 (90.4%)

|10000⟩

120 (2.9%)

|00101⟩

84 (2.1%)

|00000⟩

60 (1.5%)

|10100⟩

58 (1.4%)

|00110⟩

44 (1.1%)

|01100⟩

17 (0.4%)

|10001⟩

3 (0.1%)

X-basisHadamard rotated4096 shots

|00100⟩

1288 (31.4%)

|10000⟩

1266 (30.9%)

|00000⟩

749 (18.3%)

|10100⟩

624 (15.2%)

|10001⟩

41 (1.0%)

|10010⟩

20 (0.5%)

|10110⟩

17 (0.4%)

|00101⟩

17 (0.4%)

Y-basisS+H rotated4096 shots

|00100⟩

1371 (33.5%)

|10000⟩

1235 (30.2%)

|00000⟩

728 (17.8%)

|10100⟩

559 (13.6%)

|00110⟩

43 (1.0%)

|00010⟩

30 (0.7%)

|10010⟩

29 (0.7%)

|10001⟩

22 (0.5%)

VQE energy: -1.0946 Ha (FCI: -1.1012 Ha). Error: 4.1 kcal/mol. Outside chemical accuracy. Post-selection kept 97% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.352200
Ry(-0.352200) q[2]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-sweep-050

completed

Measured Energy

-1.0393 Ha

FCI Reference

-1.0552 Ha

Error:0.0159 Ha (9.98 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.978

<Z1>

0.978

<Z0Z1>

-1.000

<X0X1>

-0.143

<Y0Y1>

-0.122

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|00100⟩

3726 (91.0%)

|10100⟩

90 (2.2%)

|00101⟩

72 (1.8%)

|00110⟩

67 (1.6%)

|00000⟩

66 (1.6%)

|10000⟩

41 (1.0%)

|01100⟩

23 (0.6%)

|10001⟩

2 (0.0%)

X-basisHadamard rotated4096 shots

|00100⟩

1168 (28.5%)

|10000⟩

1096 (26.8%)

|00000⟩

872 (21.3%)

|10100⟩

807 (19.7%)

|00101⟩

21 (0.5%)

|10101⟩

19 (0.5%)

|10001⟩

18 (0.4%)

|00001⟩

18 (0.4%)

Y-basisS+H rotated4096 shots

|10000⟩

1114 (27.2%)

|00100⟩

1094 (26.7%)

|00000⟩

935 (22.8%)

|10100⟩

788 (19.2%)

|00010⟩

26 (0.6%)

|10010⟩

23 (0.6%)

|00110⟩

22 (0.5%)

|10001⟩

20 (0.5%)

VQE energy: -1.0393 Ha (FCI: -1.0552 Ha). Error: 10.0 kcal/mol. Outside chemical accuracy. Post-selection kept 96% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[5] q
bit[5] b

// Subspace-preserving ansatz: Ry-CNOT-X
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.143600
Ry(-0.143600) q[2]
CNOT q[2], q[4]
X q[2]

// Z-basis measurement
b = measure q

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-daemon-001

completed

Measured Energy

-1.1222 Ha

FCI Reference

-1.1373 Ha

Error:0.0151 Ha (9.45 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.972

<Z1>

0.972

<Z0Z1>

-1.000

<X0X1>

-0.170

<Y0Y1>

-0.132

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|01⟩

3338 (81.5%)

|00⟩

687 (16.8%)

|10⟩

47 (1.1%)

|11⟩

24 (0.6%)

X-basisHadamard rotated4096 shots

|10⟩

1342 (32.8%)

|00⟩

1061 (25.9%)

|01⟩

1054 (25.7%)

|11⟩

639 (15.6%)

Y-basisS+H rotated4096 shots

|10⟩

1328 (32.4%)

|00⟩

1079 (26.3%)

|01⟩

991 (24.2%)

|11⟩

698 (17.0%)

VQE energy: -1.1222 Ha (FCI: -1.1373 Ha). Error: 9.5 kcal/mol. Outside chemical accuracy. Post-selection kept 83% of Z-basis shots.

View cQASM circuit

version 3.0
qubit[2] q
bit[2] b

// Subspace-preserving ansatz: Ry-CNOT-X
// State = cos(a/2)|10> + sin(a/2)|01>, alpha=-0.223500
Ry(-0.223500) q[0]
CNOT q[0], q[1]
X q[0]

// Z-basis measurement
b = measure q

View raw JSON

IBM Torino (133q)vqe-equilibrium-002-ibm-torino

completed

Measured Energy

-1.0976 Ha

FCI Reference

-1.1373 Ha

Error:0.0397 Ha (24.91 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.879

<Z1>

0.879

<Z0Z1>

-1.000

<X0X1>

-0.412

<Y0Y1>

-0.439

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|01⟩

3739 (91.3%)

|10⟩

241 (5.9%)

|00⟩

63 (1.5%)

|11⟩

53 (1.3%)

X-basisHadamard rotated4096 shots

|10⟩

1473 (36.0%)

|01⟩

1419 (34.6%)

|11⟩

609 (14.9%)

|00⟩

595 (14.5%)

Y-basisS+H rotated4096 shots

|10⟩

1480 (36.1%)

|01⟩

1467 (35.8%)

|00⟩

591 (14.4%)

|11⟩

558 (13.6%)

View raw JSON

QI Tuna-9 (9q)vqe-equilibrium-001-tuna9

completed

Measured Energy

-1.0045 Ha

FCI Reference

-1.1373 Ha

Error:0.1328 Ha (83.3 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.723

<Z1>

0.970

<Z0Z1>

-0.709

<X0X1>

-0.060

<Y0Y1>

-0.049

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|01⟩

3483 (85.0%)

|00⟩

551 (13.5%)

|11⟩

45 (1.1%)

|10⟩

17 (0.4%)

X-basisHadamard rotated4096 shots

|10⟩

1252 (30.6%)

|00⟩

1211 (29.6%)

|01⟩

920 (22.5%)

|11⟩

713 (17.4%)

Y-basisS+H rotated4096 shots

|00⟩

1170 (28.6%)

|10⟩

1132 (27.6%)

|01⟩

1016 (24.8%)

|11⟩

778 (19.0%)

VQE on Tuna-9 achieved -1.0045 Hartree for H2 at 0.735 Å. Error of 0.133 Ha (83.3 kcal/mol) reflects significant hardware noise — Tuna-9 Bell fidelity is 87.3% vs IBM's 99%. X/Y basis correlations are heavily degraded by noise.

View raw JSON

IBM Marrakesh (156q)vqe-equilibrium-001-ibm-marrakesh

completed

Measured Energy

-1.0956 Ha

FCI Reference

-1.1373 Ha

Error:0.0417 Ha (26.1505 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.875

<Z1>

0.882

<Z0Z1>

-0.966

<X0X1>

-0.402

<Y0Y1>

-0.426

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|01⟩

3813 (93.1%)

|10⟩

213 (5.2%)

|00⟩

42 (1.0%)

|11⟩

28 (0.7%)

X-basisHadamard rotated4096 shots

|10⟩

1443 (35.2%)

|01⟩

1429 (34.9%)

|11⟩

612 (14.9%)

|00⟩

612 (14.9%)

Y-basisS+H rotated4096 shots

|10⟩

1487 (36.3%)

|01⟩

1433 (35.0%)

|11⟩

601 (14.7%)

|00⟩

575 (14.0%)

VQE on IBM Marrakesh achieved -1.0956 Hartree for H2 at 0.735 Å. Error of 0.0417 Hartree (26.2 kcal/mol) exceeds chemical accuracy (1.6 mHartree). Circuit limitations from Z-basis only prevented recovery of X/Y correlations.

View raw JSON

ibm-torinovqe-ibm-torino-rem-001

completed

Measured Energy

-1.0408 Ha

FCI Reference

-1.1373 Ha

Error:0.0965 Ha (60.54 kcal/mol)Above threshold

Energy Level Diagram

Expectation Values

<Z0>

-0.854

<Z1>

0.854

<Z0Z1>

-1.000

<X0X1>

-0.155

<Y0Y1>

-0.154

Measurement Counts by Basis

Z-basisComputational basis4096 shots

|01⟩

3218 (78.6%)

|00⟩

348 (8.5%)

|11⟩

276 (6.7%)

|10⟩

254 (6.2%)

X-basisHadamard rotated4096 shots

|10⟩

1199 (29.3%)

|01⟩

1166 (28.5%)

|00⟩

948 (23.1%)

|11⟩

783 (19.1%)

Y-basisS+H rotated4096 shots

|10⟩

1275 (31.1%)

|01⟩

1088 (26.6%)

|00⟩

927 (22.6%)

|11⟩

806 (19.7%)

View raw JSON

QI Tuna-9 (9q)vqe-tuna9-q46-001

completed

View raw JSON

Discussion

Emulator (0.75 kcal/mol): Within chemical accuracy at equilibrium and across all 14 bond distances, correctly capturing the equilibrium well and dissociation limit.

IBM Torino (0.22 kcal/mol with TREX): Chemical accuracy achieved using Qiskit's EstimatorV2 with resilience_level=1 (twirled readout error extinction). This is a 119x improvement over raw measurement (26.2 kcal/mol). Adding dynamical decoupling or gate twirling makes TREX worse -- a counterintuitive finding that "more mitigation" is not always better.

QI Tuna-9 (0.92 kcal/mol with PS+REM): Chemical accuracy achieved via a hybrid approach: post-selection catches gate leakage in Z-basis measurements, while readout error mitigation (confusion matrix inversion) corrects readout bias in X/Y-basis measurements. Validated on two independent qubit pairs: q[2,4] (0.92) and q[6,8] (1.32 kcal/mol).

IQM Garnet (14.26 kcal/mol with PS+REM): Not chemical accuracy despite very low readout error (1.1-1.3%). Gate and decoherence noise dominate on Garnet, making readout-focused mitigation insufficient. An initial attempt gave 60 kcal/mol due to an IQM-specific CNOT decomposition error: the textbook H·CZ·H decomposition fails because IQM's Hadamard differs from standard by a phase that flips X/Y correlator signs. Using R_y(π/2)·CZ·R_y(-π/2) fixed this.

Key insight: The dominant error source determines which mitigation works. IBM: readout error (TREX wins). Tuna-9: readout error (PS+REM wins). IQM Garnet: gate error (readout mitigation alone insufficient). Platform-specific gate decompositions can introduce subtle phase errors invisible to standard tests.

Sources & References

Peruzzo et al. "A variational eigenvalue solver on a photonic quantum processor" (2014)https://doi.org/10.1038/ncomms5213
Sagastizabal et al. "Experimental error mitigation via symmetry verification" (2019)https://doi.org/10.1103/PhysRevA.100.010302
Kandala et al. "Hardware-efficient variational quantum eigensolver" (2017)https://doi.org/10.1038/nature23879
O'Malley et al. "Scalable quantum simulation of molecular energies" (2016)https://doi.org/10.1103/PhysRevX.6.031007

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