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Quantum Algorithms & Circuits for Scientific Computing

Anargyros Papageorgiou Department of Computer Science Columbia University Joint work with: M. Bhaskar, S. Hadfield and I. Petras

Overview

• Why quantum algorithms for scientific computing
• Requirements
• Quantum algorithms & circuits for fundamental

functions, e.g., 𝑥, ln 𝑥 etc.

– Algorithms by combining elementary modules – Applications – Tests

• Summary

Why quantum circuits for scientific computing

• Scientific computing applications can benefit from fast quantum

algorithms

– e.g., numerical linear algebra problems

• Typically we need to compute fundamental functions such as

𝑦, ln 𝑦 , sin 𝑦

• Classical computation: IEEE standard for floating point arithmetic

– Comprehensive math. libraries

• Quantum computation:

– No standard for numerical computations – No general purpose quantum circuits implementing fundamental functions – Details about numerical calculations have been avoided in a way

Requirements – quantum circuit model

• Develop a standard for numerical computations

– Reversible computation

• Fixed precision representation of numbers
• Quantum algorithms for scientific computing with

performance guarantees

– error & cost

Quantum algorithms &circuits for fundamental functions

• Library of elementary quantum circuit templates

implementing

– Arithmetic expressions – Shifts – Initial approximations for iterative methods – New circuits are added to the library as they are derived

• Elementary quantum circuits are modules with known

error and cost characteristics

• New algorithms are implemented by combining

modules

Algorithms Input: 𝑜+1 qubit register

𝑥 = 𝑥𝑡 ⊗ 𝑥(𝑛−1) ⊗ ⋯ ⊗ 𝑥(0) ⊗ 𝑥 −1 ⊗ ⋯ ⊗ |𝑥(𝑛−𝑜)⟩ sign integer part fractional part 𝑥𝑡, 𝑥(𝑘) ∈ {0,1}, 𝑥 = −1 𝑥𝑡 𝑥 𝑘 2𝑘

𝑛−1 𝑘=𝑛−𝑜

• Register length may be different for intermediate calculations.

Elementary quantum circuit template examples

1.

• Addition and multiplication imply that the format of 𝑠𝑓𝑡 is

known given the format of the inputs

• 𝑜 bit inputs: 𝑠𝑓𝑡 can be represented exactly using 2𝑜 + 1 bits

(qubits) (plus sign) of which 2(𝑜 − 𝑛) hold the fractional part

• Any desired number 𝑐 of significant bits after the decimal

point in 𝑠𝑓𝑡 can be passed on to the next stage

|𝑦⟩ |𝑧⟩ |𝑨⟩ 𝑠𝑓𝑡 = 𝑦𝑧 + 𝑨 |𝑧⟩ |𝑨⟩ |𝑠𝑓𝑡⟩

2.

Quantum circuit which on input 𝑥 ≥ 1 computes 𝑦 0 = 2−𝑞 for 𝑞 such that 2𝑞 > 𝑥 ≥ 2𝑞−1. Initial approximation for Newton iteration computing 𝑥−1, 𝑥 ≥ 1 Other similar circuits are also included

Applications

We have derived quantum algorithms for:

• 𝑥−1
• sin 𝑥 , cos 𝑥
• Inverse trigonometric
• 𝑥
• 𝑥1/2𝑗, 𝑗 = 1, … , 𝑙
• ln 𝑥
• 𝑥𝑔, 𝑔 ∈ 0,1

Earlier work [Cao, P, Petras, Traub, Kais] Poisson equation

Square root - 𝑥, w ≥ 1 Use Newton iteration Selection of function whose zero is 𝑥 is important

e.g., 𝑔 𝑦 = 𝑦2 − 𝑥 is not a good choice Iterative step 𝑦𝑗 = 𝑦𝑗−1 − (𝑦𝑗−1

2

− 𝑥)/(2𝑦𝑗−1) requires division Need extra circuitry to keep track of location of decimal point in result

We use:

• 1. One iteration 𝑦𝑗 → 𝑥−1

𝑦𝑗 = 𝑕1 𝑦𝑗−1 ≔ −𝑥 𝑦𝑗−1

2

+ 2 𝑦𝑗−1, 𝑗 = 1, … , 𝑡1

• 2. Second iteration 𝑧𝑘 →

1 𝑦𝑡1 ≈

𝑥 𝑧𝑘 = 𝑕2 𝑧𝑘−1 ≔ 3 𝑧𝑘−1 − 𝑦𝑡1 𝑧𝑘−1

3

2 , 𝑘 = 1, … , 𝑡2

Quantum circuit for 𝑥,

• 𝑧

0, 𝑦 0 init. approx.

• In each stage calculations are exact
• Results are truncated to 𝑐 bits (qubits) after the decimal point

and passed on to the next stage

.

𝑥 = 𝑛 bits 𝑜 − 𝑛 bits

Thm. 𝑧 𝑡 − 𝑥 ≤ 3 4

𝑐−2𝑛

2 + 𝑐 + log2 𝑐 , 𝑐 ≥ max 2𝑛, 4 Cost:

• # iterative steps: 𝑡 ≔ 𝑡1 = 𝑡2 = 𝑃(log2 𝑐)
• # qubits per step: 𝑃 𝑜 + 𝑐
• # quantum ops. per step: low degree poly in 𝑜 + 𝑐

Logarithm - ln 𝑥 , 𝑥 > 1 Algorithm:

• 1. Shift right 𝑥 to obtain 𝑥𝑞 = 21−𝑞𝑥 ∈ (1,2), with

2𝑞 > 𝑥 ≥ 2𝑞−1

• 2. Compute 𝑢𝑞 = 𝑥𝑞

1/2ℓ. Note 𝑢𝑞 = 1 + 𝜀, with 𝜀 ≈ 2−ℓ

• 3. Approximate ln 𝑢𝑞 ≈ 𝜀 −

𝜀2 2

• 4. ln 𝑥 ≈ (𝑞 − 1) ln 2 + 2ℓ 𝜀 −

𝜀2 2

Quantum circuit for ln 𝑥 𝑧𝑞 = 𝑔 𝑢𝑞 = 𝑢𝑞 − 1 −

𝑢𝑞−1

2

2

≈ ln 𝑢𝑞 (step 3 of alg.) 𝑨𝑞 = 2ℓ𝑧𝑞 (step 4 of alg.)

Thm.

[ 𝑞 − 1 ln 2 + 𝑨𝑞] − ln 𝑥 ≤ 3 4

5ℓ/2

𝑛 + 32 9 + 2 32 9 + 𝑜 ln 2

3

where ℓ ≥ log2 8𝑜 , 𝑐 ≥ max 5ℓ, 25 Cost: Total # qubits is proportional to ℓ 𝑜 + 𝑐 log2𝑐 Total # of quantum operations is proportional to ℓ times a poly in 𝑜 + 𝑐

Tests

𝑥: Comparison between our algorithm and Matlab

ln 𝑥: Comparison between our algorithm and Matlab

Summary

• Quantum algorithms & circuits for

fundamental functions

• Performance guarantees (accuracy, cost)
• Modular design
• Easy to derive error bounds
• Easy to derive resource estimates
• Internal implementation details can be changed

transparently