Table 1 T-count achieved by different methods on a set of benchmark circuits

From: Quantum circuit optimization with AlphaTensor

Circuit

Number of qubits

T-count without gadgets

T-count with gadgets

Original

Compiled

Baselines

AlphaTensor-Quantum

Original

AlphaTensor-Quantum

8-bit adder

24

⋆

129 (ref. 17)

139

114 (57Tof)

94 (33Tof + 28T)

Barenco Toff3

5

8

13 (refs. 23,24)

13b

8 (4Tof)

4 (2Tof)

Barenco Toff4

7

14

24 (refs. 17,23)

23

16 (8Tof)

8 (4Tof)

Barenco Toff5

9

20

34 (refs. 17,23)

33

24 (12Tof)

12 (6Tof)

Barenco Toff10

19

50

84 (ref. 17)

83

64 (32Tof)

32 (16Tof)

CSLA-MUX3

15

21

40 (ref. 23)

39

20 (10Tof)

16 (8Tof)

CSUM-MUX9

30

42

72 (refs. 17,20)

71

56 (28Tof)

28 (14Tof)

GF(22)-multc

6

6

17a

17b

8 (4Tof)

6 (3Tof)

GF(23)-multc

9

9

31a

29

18 (9Tof)

12 (6Tof)

GF(24)-mult

12

12

47 (ref. 23)

39

32 (16Tof)

18 (9Tof)

GF(25)-mult

15

15

84 (ref. 23)

59

50 (25Tof)

26 (13Tof)

GF(26)-mult

18

18

118 (ref. 23)

77

72 (36Tof)

36 (18Tof)

GF(27)-mult

21

21

167 (ref. 24)

104

98 (49Tof)

44 (22Tof)

GF(28)-mult

24

24

214 (ref. 20)

123

128 (64Tof)

58 (29Tof)

GF(29)-mult

27

27

295 (ref. 17)

161

162 (81Tof)

70 (35Tof)

GF(210)-mult

30

30

350 (ref. 17)

196

200 (100Tof)

92 (46Tof)

Grover5

9

⋆

44 (ref. 17)

152

96 (48Tof)

66 (27Tof + 12T)

Hamming15 (high)

20

⋆

787a (1,010 (ref. 17))

773

702 (351Tof)

440 (173Tof + 2CS+ 90T)

Hamming15 (low)

17

42

75 (ref. 17)

73

46 (23Tof)

34 (17Tof)

Hamming15 (med)

17

⋆

156a (162 (ref. 17))

156

164 (82Tof)

78 (35Tof + 8T)

HWB6

7

27

51 (ref. 17)

51

30 (15Tof)

20 (10Tof)

Mod-Adder1024

28

⋆

798a (978 (ref. 17))

762

570 (285Tof)

500 (141Tof + 15CS+ 188T)

Mod-Mult55

9

11

17 (ref. 17)

17b

14 (7Tof)

6 (3Tof)

Mod-Red21

11

28

55 (refs. 17,23)

51

34 (17Tof)

22 (11Tof)

Mod 54

5

5

7 (refs. 20,23,24)

7b

8 (4Tof)

2 (1Tof)

QCLA-Adder10

36

⋆

116 (ref. 17)

135

68 (34Tof)

94 (28Tof + 5CS+ 28T)

QCLA-Com7

24

42

59 (ref. 17)

59

58 (29Tof)

24 (12Tof)

QCLA-Mod7

26

⋆

165 (ref. 17)

199

118 (59Tof)

122 (43Tof + 36T)

QFT4

5

43

55 (ref. 17)

53

59 (2Tof + 55T)

44 (4Tof + 3CS+ 30T)

RC-Adder6

14

24

37 (refs. 17,23)

37

22 (11Tof)

12 (6Tof)

NC Toff3

5

7

13 (refs. 17,23,24)

13b

6 (3Tof)

4 (2Tof)

NC Toff4

7

11

19 (refs. 17,23,24)

19b

10 (5Tof)

6 (3Tof)

NC Toff5

9

15

25 (ref. 17)

25

14 (7Tof)

8 (4Tof)

NC Toff10

19

35

55 (ref. 17)

55

34 (17Tof)

18 (9Tof)

VBE-Adder3

10

14

20 (refs. 17,23,24)

19b

20 (10Tof)

6 (3Tof)

  1. Numbers in bold indicate the best T-count for each case (with or without gadgets). Even without gadgetization, AlphaTensor-Quantum matches or outperforms the considered baselines for all circuits that have not been split into sub-circuits (split circuits are marked with a star). When considering gadgets, AlphaTensor-Quantum further reduces the T-count drastically and generally outperforms the original constructions with gadgets. The notation ‘aTof + bCS + cT’ indicates a circuit with a Toffoli gates, b CS gates and c T gates (when multiple solutions from AlphaTensor-Quantum achieve the same T-count but differ in the trade-off between gadgets and T gates, we report the result with the highest number of Toffoli gates). aBaseline results were obtained with the methods in Supplementary Section B and are better than the best-published T-count (provided in parentheses where applicable). bResults were proven to be optimal decompositions of the given tensor using the Z3 theorem prover. cThe GF(22)-mult and GF(23)-mult circuits are included for completeness (Supplementary Section D.1).
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