1. OSU Quantum Information Seminar
Introduction to Quirk
Daniel Gauthier
Department of Physics
OSU Quantum Information Seminar
September 21, 2018

2. Future of computing
GPU
We are still in the “nuts and bolts” stage, but rapid progress in the past couple of years
CPU
QPU
Quantum internet
Quantum Processor Unit
The quantum processor will be best suited to speed up “hard” problems
Software is being developed to make accessing a QPU easier (e.g., Rigetti mixed classical and noisy quantum processors)

3. Outline
Physical qubits
IBM quantum experience
quirk
Single-qubit gates
Two-qubit gates
Classical Fourier transform
Quantum Fourier transform

4. Simple quantum picture of an atom
Bohr’s model of the hydrogen atom postulates that the electron behaves as a wave (de Broglie wave) and that an integral number m of waves must fit in a close orbit
Quantized angular momentum
And energy!
Hydrogen: 1 proton + 1 electron

5. Discrete energy “levels”
m = 3
Quantum jumps between energy states go hand-in-hand with photon emission or absorption
m = 2
m = 1
energy
T1 time

6. Matter qubit “any” energy eigenstates can be used to realize a qubit
Can induce transitions or create superposition states by shining resonate light on the atom
The sharpness of the levels allows us to ignore all of the other energy states
energy
a and b are complex probability amplitudes

7. A graphical method for describing the state of a qubit
Bloch sphere
A qubit is described by a complex vector space!
T2 time describes the time scale for decay of superposition state
A qubit state requires specifying two probability amplitudes a and b, which are complex. But there is a normalization condition, so requires 2(21-1)=2 independent real numbers

8. Interpretation of points on the Bloch sphere
Atom is in the ground state
Atom is in a superposition state
Atom is in the exited state
Errors: “z” errors, “x” errors (or “xy” errors)

9. Outline
Physical qubits
IBM quantum experience
quirk
Single-qubit gates
Two-qubit gates
Classical Fourier transform
Quantum Fourier transform

10. Noisy quantum computers can be accessed over the web
IBM Quantum Experience

11. Quantifying noisy
IBM Quantum Experience

12. Quantum Composer
IBM Quantum Experience

13. Outline
Physical qubits
IBM quantum experience
quirk
Single-qubit gates
Two-qubit gates
Classical Fourier transform
Quantum Fourier transform

14. Quirk
Quirk is similar to the IBM Quantum Composer, but is a bit easier to use for a first-time user wanting to explore gate-level programming of a quantum computer
(search on “quirk quantum”)
Written by a quantum information scientist in his “spare time,” now is a Google engineer

15. Quirk is a classical simulation tool
The back-end of quirk evaluations the operation of the quantum circuit using the rules of quantum mechanics using a classical computer
There will be no “speed up” because we are evaluating quantum theory on a classical computer, but you can start to learn how a quantum computer operates at the gate level (think of machine language programming in the early days of classical computing, assembly language, etc.)

16. Limitations of Quirk
In quantum theory, a many qubit system is represented by a state that is a tensor product of the individual qubit states
Specifying the state of an n-qubit system requires a complex vector that is size 2n (but the normalization condition reduces this somewhat). Need 2(2n-1) independent real numbers to specify the state (even though there are only n physical qubits)
220 = 1.0×106
225 = 3.4×107
250 = 1.1×1015
Quirk can handle ~20-30 qubits depending on the problem

17. Two qubits

18. Two qubits (n=2)

19. Vector/matrix representation of quantum states
Using vector/matrix representation of quantum states
Matrix form
Start in state as Quirk assumes (and IBM machine attempts to initialize in this state)

20. Outline
Physical qubits
IBM quantum experience
quirk
Single-qubit gates
Two-qubit gates
Classical Fourier transform
Quantum Fourier transform

21. A “Hamiltonian” is use to describe qubit operation
In quantum theory, the effect of a gate operation (manipulation of one or more qubits) is described mathematically by a Hamiltonian. In gate-based quantum computing, we can write a general quantum algorithm in terms of single- and two-qubit gates.
A physically realizable Hamiltonian appropriate for quantum operations will have the form of a 2n × 2n Hermitian matrix with 22n independent real numbers. The matrix is also unitary.

22. Creating a superposition of a single qubit
Single qubit (n=1) gate known as the Hadamard operator creates a superposition state. Requires a 21 × 21 Hermitian matrix
Start in state
Output state

23. Hadamard gate in Quirk

24. Quantum NOT gate (Pauli-X gate)
Matrix form

25. Outline
Physical qubits
IBM quantum experience
quirk
Single-qubit gates
Two-qubit gates
Classical Fourier transform
Quantum Fourier transform

26. Two qubit system (n=2)
The state of the system is described in terms of the tensor product of the two individual states
We need 22=4 complex probability amplitudes. Using the “computational” basis states, find
Base-10 (decimal)

27. n-qubit gates
An n-qubit gate is described by a 2n× 2n Hermitian matrix
Requires a lot of multiplications when n is large!

28. Controlled-NOT two-qubit gate
Controlled NOT: state of “target” qubit depends on state of “control” qubit
Matrix form

29. Creating an entangled state
Output state can’t be written as the product of the two qubit states

30. Creating an entangled state: matrix representation

31. CNOT gate in quirk, making an entangled state

32. Outline
Physical qubits
IBM quantum experience
quirk
Single-qubit gates
Two-qubit gates
Classical Fourier transform
Quantum Fourier transform

33. Classical Discrete Fourier Transform
Linear transformation of a set of N complex numbers xa {x0, x1, …, xN-1}
time/frequency, used to find the “spectrum” of a set of time series data
N2 multiplications (Fast Fourier Transform requires (N log2N) multiplications)

34. Classical Discrete Fourier Transform: Matrix Form
Linear transformation of a set of N complex numbers xa {x0, x1, …, xN-1}
Unitary matrix
N×N matrix
N×1 vector

35. Classical Discrete Fourier Transform: Matrix Form
Linear transformation of a set of N complex numbers xa {x0, x1, …, xN-1}
Unitary matrix
N×N matrix
Quantum concept:
Use n qubits to represent the vector, where N=2n or n=log2N
Classical simulation still takes N memory elements and an N×N matrix.
N×1 vector

36. Outline
Physical qubits
IBM quantum experience
quirk
Single-qubit gates
Two-qubit gates
Classical Fourier transform
Quantum Fourier transform

37. Quantum Fourier Transform
Use decimal short-hand notation for the quantum states
If the initial states are computational basis states (xa = 0 or 1), can just initialize the input qubits in either the state or
Factoring integers via Shor’s algorithm requires a method for finding the period of a sequence of numbers, which only requires a quantum Fourier transform using these basis states

38. Quantum Fourier Transform
Use decimal short-hand notation for the quantum states
If the initial states are computational basis states (xa = 0 or 1), can just initialize the input qubits in either the state or
Factoring integers via Shor’s algorithm requires a method for finding the period of a sequence of numbers, which only requires a quantum Fourier transform using these basis states
(Have a very long sequence of binary numbers and need to find when a pattern repeats in the sequence)

39. Quantum Fourier Transform for 3 qubits (n=3)
Decimal notation
Binary notation
These quantum states require 23 = 8 complex numbers to specify
Or 23 = 8 complex numbers are represented by these states
is represented by an 8 × 8 matrix
How do we realized using single- and two-qubit gates?

40. Phase gate
Circuit requires a Hadamard gate and a controlled “phase gate”
Single-qubit version of phase gate (operation on target gate):
Rotate the state about the z-axis by p
Rotate the state about the z-axis by p/2
Rotate the state about the z-axis by p/4

41. QFT circuit for n=3
Requires n(n+1)/2 gates (there are more efficient circuits)
Performs the Fourier transform for 23 = 8 classical bits
Output state of circuit
SWAP gate

42. QFT Quirk (n=3)

43. Summary
Quirk is an easy-to-use simulation tool for gate-based quantum computation
The quantum Fourier transform has an exponential speed up in comparison to its classical counterpart because
n qubits represent a N=2n sequence of binary numbers
the QFT only requires n(n+1)/2 gates where as the classical FT requires N2 multiplications (or N log2N for the FFT)