Notes on: Ekert, A., Hayden, P., & Inamori, H. (2000): Basic concepts in quantum computation

Table of Contents

Notation

  • ekert00_basic_concep_quant_comput_1f89c145b527cc5f951730ad58723c6493dbad83.png stands for ekert00_basic_concep_quant_comput_2e4e81895a034bae2d39bd11a4dbd8f07ff06bce.png where ekert00_basic_concep_quant_comput_e1ca66ef71bcf995b977831f0e99d8dd4be564d7.png, e.g. ekert00_basic_concep_quant_comput_efd12937fc013388c000cc35d85e3a8cf9d514b7.png

Definitions

The Hadamard gate is a single qubit gate ekert00_basic_concep_quant_comput_f1330523c697c7f312e3dcdd0386cfbf0532a867.png performing the unitary transformation known as the Hadamard transform is defined as

ekert00_basic_concep_quant_comput_f9de2963710e0526e4edd6432edf470713140792.png

or

ekert00_basic_concep_quant_comput_fd19523824ab3db26a69314f4c8b45a418fe68d4.png

The phase shift gate a single qubit gate ekert00_basic_concep_quant_comput_2a275ba57b65332c8eb71ed5f6713abb5db0e66b.png defined as:

ekert00_basic_concep_quant_comput_29b7cccaa539c576732980ceaf8f1cd12f907c53.png

or in matrix-representation:

ekert00_basic_concep_quant_comput_512c3e9bdd3d794dacd04ca5f1bb809664111742.png

The controlled NOT gate (C-NOT) flips the second (target) qubit if the first (control) qubit is ekert00_basic_concep_quant_comput_a19ddbea12905415f9d79dd893b28c48c03c6255.png and does nothing if the control qubit is ekert00_basic_concep_quant_comput_59ed23b99ac1f420709805adb81c9e5e3bf3badc.png.

The gate is represented by the unitary matrix

ekert00_basic_concep_quant_comput_8419db2ea0d6c596ee0b20bf8c98d8e605eeb647.png

or

ekert00_basic_concep_quant_comput_1c30eaf16d4aa1bcb5778b1dd821701abf44a0f3.png

and ekert00_basic_concep_quant_comput_cc0337454ae489f34d65d57561bd890104d1dedd.png denotes the XOR or addition (or more precisely, direct sum) modulo 2.

The phase shift gate ekert00_basic_concep_quant_comput_eb67e089bd4009be62cbb54ab3910ba427c436dc.png is a two-qubit state defined my

ekert00_basic_concep_quant_comput_1d9ef61859b1d908cdbbc2fd8415073acb1549d7.png

or

ekert00_basic_concep_quant_comput_f1f1450880e7ffc69c360d3a8212b5bbfad59683.png