1. Chap 4 Quantum Circuits: p 171-215
Dr. Charles Tappert
The information presented here, although greatly condensed, comes almost entirely from the course textbook: Quantum Computation and Quantum Information by Nielsen & Chuang

2. 4 Quantum Circuits
This chapter develops the fundamental principles of quantum computation
Establishes the building blocks for quantum circuits, a universal language for describing sophisticated quantum computations
The next two chapters cover the two known-to-date fundamental quantum algorithms
Quantum Fourier transform
Quantum search algorithm

3. 4 Quantum Circuits This chapter introduces two main ideas
Quantum circuit model
The fundamental model of quantum computation
Universal sets of gates
That can express any quantum computation
This chapter is the most reader-intensive of all the chapters, with high density of exercises
Facility with the quantum circuit model is easy but requires grasping a large number of simple results and techniques

4. 4.1 Quantum Algorithms
The spectacular promise of quantum computers is to enable new algorithms, which allows solving problems requiring huge resources on a classical computer
Figure The main quantum algorithms and their relationships and notable applications
Note quantum counting algorithm, a combination of quantum Fourier transform and search algorithms

5. 4.1 Quantum Algorithms

6. 4.2 Single Qubit Operations
Common Single Qubit Gates

7. 4.2 Single Qubit Operations
The single qubit state can be visualized as a point on the unit sphere
called the Bloch sphere representation
And the Bloch vector is

8. 4.2 Single Qubit Operations
Exercise 2.11 required by Exercise 4.1

9. 4.2 Single Qubit Operations
Exercise 4.1

10. 4.2 Single Qubit Operations
Rotation operators for the Pauli matrices

11. 4.2 Single Qubit Operations
Exercise 4.2

12. 4.2 Single Qubit Operations
Exercise 4.7

13. 4.2 Single Qubit Operations
An arbitrary unitary operator on a single qubit can be written in many ways as a combination of rotations and global phase shifts
Theorem 4.1: (Z-Y decomposition for a single qubit) Suppose U is a unitary operation on a single qubit. Then there exist real numbers
proof in textbook

14. 4.2 Single Qubit Operations
The utility of Theorem 4.1 is in the corollary
Corollary 4.2: Suppose U is a unitary gate on a single qubit. Then there exist unitary operators A, B, C on a single qubit such that

15. 4.2 Single Qubit Operations
Exercise 4.13

16. 4.2 Single Qubit Operations
Common Single Qubit Gates

17. 4.3 Controlled Operations
The controlled operation is one of the most useful in computing, classical and quantum
If A is true, then do B
Basic CNOT operation More general controlled U X=

18. 4.3 Controlled Operations
More CNOT gate information from the book Approaching Quantum Computing click here
Computes CNOT information
Using ket/bra and matrix/vector formulations
In particular, computes CNOT matrix as the sum of the outer products of the input and output basis bra/ket vectors

19. 4.3 Controlled Operations Controlled-U with Single Qubit Ops & CNOT
Understand how to implement controlled-U operation for arbitrary single qubit U using only single qubit operations and the CNOT gate
Step one of two-part procedure
Apply phase shift exp(ia) on the target qubit controlled by the control qubit
That is

20. 4.3 Controlled Operations Controlled-U with Single Qubit Ops & CNOT
Verify right hand side works

21. 4.3 Controlled Operations Controlled-U with Single Qubit Ops & CNOT

22. 4.3 Controlled Operations Controlled-U with Single Qubit Ops & CNOT
Step two of two-part procedure – figure 4.6
From Corollary 4.2, let where A, B, and C are single qubit ops and ABC = I
If the control qubit is set, then is applied to the second qubit, otherwise the operation ABC = I is applied yielding no change
Thus, the right hand side works and it contains only CNOT and single qubit operations

23. 4.3 Controlled Operations Controlled-U with Single Qubit Ops & CNOTX X

24. 4.4 Measurement
To denote a projective measurement in the computational basis, we use the meter symbol

25. 4.4 Measurement
There are two important principles about quantum circuits worth emphasis
Principle of deferred measurement: measurements can always be moved to the end of the circuit
If measurement results are used in the circuit, classically controlled operations can be replaced by quantum controlled operations
Principle of implicit measurement: quantum wires not measured at the end of the circuit can be assumed to be measured
Obvious but surprisingly useful result

26. 4.5 Universal Quantum Gates
Universal classical computation gates exist
Because the Toffoli gate is universal for classical computation, quantum circuits subsume classical circuits
Similarly, there exist sets of gates universal for quantum computation

27. 4.5.1 Two-level Unitary Gates are Universal
An arbitrary unitary matrix U can be decomposed into a product of two-level unitary matrices
Matrices acting non-trivially only on two-or-fewer vector components
Proof in textbook

28. 4.5.2 Single Qubit and CNOT Gates are Universal
The previous section showed that an arbitrary unitary matrix U can be decomposed into a product of two-level unitary matrices
We now show that single qubit and CNOT gates together can implement an arbitrary two-level unitary operation on the state space of n qubits
Therefore, single qubit and CNOT gates are universal

29. 4.5.3 A Discrete Set of Universal Operations
The previous section showed that single qubit and CNOT gates are universal
But no method is known to implement these gates so they are resistant to errors
We now find a universal set of gates that chapter 10 will show perform in error-resistant fashion using quantum error-correcting codes

30. 4.5.3 A Discrete Set of Universal Operations
The previous section showed that single qubit and CNOT gates are universal
But no method is known to implement these gates so they are resistant to errors
We now find a universal set of gates that chapter 10 will show perform in error-resistant fashion using quantum error-correcting codes

31. 4.5.4 Approximating Arbitrary Unitary Gates is Generally Hard
We have seen that any arbitrary transformation on n qubits can be built from a small set of elementary gates
However, this it is often not possible to do this efficiently
Most can only be implemented very inefficiently

32. 4.5.4 Approximating Arbitrary Unitary Gates is Generally Hard
Number of patches increases doubly exponentially in the number of qubits n

33. 4.6 Summary of the Quantum Circuit Model of Computation
In this book the term quantum computer = quantum circuit model of computation
Summary of key elements of the model
Classical resources: classical + quantum parts
A suitable state space: 2n-dimensional Hilbert space
Ability to prepare states in the computational basis
Ability to perform quantum gates
Ability to perform measurements in the computational basis

34. 4.7 Simulation of Quantum Systems
A most important practical application of computation is simulation of physical systems
Engineering design of buildings, automobiles, etc.
This section presents a quantum algorithm for simulation
And an illustrative example