Quantum Computing, Rieffel & Polak Chapter 5

1. Quantum Computing, Rieffel& PolakChapter 5 Drs. Charles Tappert and Ron Frank The information presented here, although greatly condensed, comes almost entirely from the textbook

2. Chapter 5 – Quantum State Transforms • This chapter develops the basic mechanisms for computing on quantum information • Sec 5.1 discusses unitary transformations and no cloning • Sec 5.2 covers one- and two-qubit transformations • Sec 5.3 describes applications to two problems – dense coding and quantum state teleportation • Sec 5.4 shows that any transform can be represented as a sequence of one- and two-qubit transforms • Sec 5.5 discusses finite sets of gates to approximate all quantum transforms universally • Sec 5.6 describes the standard circuit model

3. Chapter 5 – Quantum State Transforms5.1 Unitary Transformations • A transform is a mapping from state space to itself • The transformations must be linear • The transformations must be unitary • Unitary operators • Preserve inner product • Map orthonormal bases to orthonormal bases • Columns/rows of the matrix form are orthonormal • Operators are reversible

4. Chapter 5 – Quantum State Transforms5.1 Unitary Transformations • No-Cloning Principle – proof by contradiction • Suppose the unitary transform U clones, then • Proof by contradiction

5. Chapter 5 – Quantum State Transforms5.2 Some Simple Quantum Gates • Quantum gate = transform acting on small # qubits • Quantum circuit = sequence of quantum gates • Sample graph of three-qubit quantum circuit Each line represents a qubit

6. Chapter 5 – Quantum State Transforms5.2 Some Simple Quantum Gates • Pauli gates = most common single-qubit gates Identity operation Not operation Y = ZX = not + phase change Changes relative phase

7. Chapter 5 – Quantum State Transforms5.2 Some Simple Quantum Gates • The Hadamard Gate or Transformation

8. Chapter 5 – Quantum State Transforms5.2 Some Simple Quantum Gates • Multiple-qubit gates from single-qubit gates • Some can be created as tensor products of single-qubit gates – these are not very interesting • More interesting are those that can change the entanglement between qubits of the system

9. Chapter 5 – Quantum State Transforms5.2 Some Simple Quantum Gates • Controlled-NOT & other singly controlled gates

10. Chapter 5 – Quantum State Transforms5.2 Some Simple Quantum Gates • Controlled-NOT • Flips 2nd bit if 1st is 1, otherwise leaves it unchanged • It is unitary and its own inverse • Cannot be decomposed into a tensor product of two single-qubit transformations • Can take unentangled 2-qubit to entangled 2-qubit • Or vice versa, entangled to unentangled • So common it has its own graphical notation Open circle indicates control bit X indicates negation of target bit

11. Chapter 5 – Quantum State Transforms5.2 Some Simple Quantum Gates • C-NOT generalizes to controlled any 2-qubit gate

12. Chapter 5 – Quantum State Transforms5.2 Some Simple Quantum Gates • Graphical icons can be used to create circuits • For example, the swap circuit

13. Chapter 5 – Quantum State Transforms5.3 Applications of Simple Gates • We will discuss dense coding and teleportation • The key to both is using entangled particles • The initial setup is the same for both apps • Alice and Bob are each sent one of an EPR pair

14. Chapter 5 – Quantum State Transforms5.3 Applications of Simple Gates • Alice can perform transforms on her particle and Bob on his, until Alice sends Bob her particle or Bob sends Alice his particle • Alice can perform transforms only of the form • Bob can perform transforms only of the form

15. Chapter 5 – Quantum State Transforms5.3 Applications of Simple Gates

16. Chapter 5 – Quantum State Transforms5.3 Applications of Simple Gates • Alice can transmit 2 classical bits via 1 qubit • Alice performs this encoding on her qubit

17. Chapter 5 – Quantum State Transforms5.3 Applications of Simple Gates • Bob decodes the information Bob measures the 2 qubits to obtain to obtain the 2-bit code sent by Alice

18. Chapter 5 – Quantum State Transforms5.3 Applications of Simple Gates

19. Chapter 5 – Quantum State Transforms5.3 Applications of Simple Gates • Quantum teleportation transmits enough classical bit information about the quantum state of a particle so a receiver can reconstruct the exact quantum state of the particle • The decoding step is the encoding step of dense coding, and the encoding step is the decoding step of dense coding • See this section of book for details

20. Chapter 5 – Quantum State Transforms5.4 Realizing Transforms as Circuits • Arbitrary unitary quantum transforms can be implemented from a set of primitive transforms • The two-qubit C-NOT plus three types of single-qubit gates • The intuitive idea behind this is that any unitary transform is simply a rotation of the complex vector space underlying the n-qubit quantum state space

21. Chapter 5 – Quantum State Transforms5.5 Universally Approximating Gate Set • Sec 5.4 showed all transforms can be realized by a sequence of one-qubit and C-NOT gates • But we want to deal with a finite set of gates • This section shows that a finite set of gates can approximate all unitary quantum transforms • The finite set is

22. Chapter 5 – Quantum State Transforms5.6 The Standard Circuit Model • The standard circuit model describes all computations in terms of a circuit composed of simple gates followed by a sequence of measurements • Single-qubit gates and two-qubit C-NOT gates • Single-qubit measurements in the standard basis • For clarity, the n qubits are often organized into registers, subsets of the n qubits