Quantum pricing with a smile: implementation of local volatility model on quantum computer

EPJ Quantum Technology

Table 1 Resources for the elementary gates. We here assume that operands are n-bit. We omit subleading terms with respect to n

Gate	Logical qubits				T-count	Reference
Gate	Total	Operand	Output	Ancilla	T-count	Reference
Adder	2n	2n	0 (self-update)	0	14n	[37, 38, 46]
Ctrl Adder	2n	2n	0 (self-update)	0	21n	[42, 46]
Modular Adder^∗	2n	2n	0 (self-update)	0	70n	[26, 37, 38, 46]
Multiplier	3n	2n	n	0	\(21n^{2}\)	[42]
Divider	5n	2n	n	2n	\(35n^{2}\)	[46]
Multi Ctrl Toffoli	2n	n	1	n	8n	[18, 48]
Square Root^†	4n	n	n	2n	\(14n^{2}\)	[52]
arccos	105				3.4 × 10⁴	[51]
controlled rotation (with accuracy of \(2^{-n}\))	2				3n	[16, 47, 53]

^∗Since the modular adder is constructed by 5 plain adders [26], its T-count is 5 times the values of the adder.
^†The circuit given in [52] takes an n-bit input and returns an n/2-bit output of square root and an n/2-bit remainder. To keep n-bit accuracy, we add n qubits with 0’s to the input of the circuit and calculate the n-bit square root with 2n-bit input. The added n bits are treated as the input, and the n bits remainder is treated as ancillae.