1. Engineering Models and Design Methods for Quantum State Machines.

2. 1. Synthesis Flow for Quantum State Machines (QSM) – Models of QSM – Models of Controller – High Level Synthesis – Low Level Synthesis DCARL 2. Examples

3. 1.2 What is this lecture about? Objectives: 1.Develop models for Quantum State Machines (QSMs) for analysis and synthesis. 2.Develop models for a classical controller which controls the QSM. 3.Develop a step by step procedure for going from abstract specifications (state graphs, signal transition graphs, state tables) to the gate level quantum circuit. 4.Develop a software to automate the low level logic synthesis step.

4. 2. Basics of qubits, quantum gates etc. Qubit – Basic unit of quantum information, analogous to “bit” in classical logic. – Unlike a “bit” which has to have a value that is either ‘0’ or ‘1’, a “qubit” can exist in a superposition of basis states. – Represented as |Ψ>= α |0> +β|1> Where α, β are probability amplitudes such that |α | 2 +|β| 2 =1. |0> and |1> are called the basis states. – Any two level system can represent a qubit. Eg: Electrons with spin +1/2 and -1/2; photons with horizontal and vertical polarization etc.

5. Reminder about quantum gates and circuits Quantum gates – A logical operation performed on a qubit. – Usually a series of control pulses (eg: lasers, microwaves) used to control a qubit. – The gate exists only for the duration of the control pulses! Very different from classical logic. – Represented as a unitary matrix. – Gates in quantum are reversible unlike their classical counterparts. Quantum Array – A series of logical operations (i.e. gates) applied to a qubit. – Gates are applied sequentially since no two pulses can simultaneously be applied on a qubit.

6. Quantum Gate operates in time not in space A universal gate for quantum operations – Toffoli gate. – Performs Controlled-Controlled-NOT operation. – The qubit ‘c’ flips if both a and b are 1. i.e. C next =ab c Note: When c=1, Nand operation is performed on ab!! Illustration: C abcabc 1.Explain in detail how quantum gate operates in time, based on controlled pulses. 2.Classical computer creates quantum pulses, such as EM

7. Quantum Computing requires classical computer for operation General concept of Quantum computing: 1.A classical computer “initializes” the qubits from a stream of bits. 2.“Computation” is performed (i.e. unitary transformations applied on qubits) 3.The classical computer performs a “measurement” (read out). INITIALIZATION COMPUTATION MEASUREMENT time CLASSICAL COMPUTER WORKS AS A CONTROLER

8. 3. Synthesis Flow of designing quantum state machines

9. 3.1 Models of QSM 1.Existing models like Quantum Turing Machines (QTM) are very mathematical or abstract. 2.We need practical models for implementation. 3.Three models are proposed in this thesis.

10. QSMs with Classical Memory (QSM-CM)

11. 1.Model I: QSMs with Classical Memory (QSM-CM). External memory required. Measurement is performed after each pass through the quantum array (QA). Re-initialization required before each pass through the QA. Very similar to classical logic. Output bits Classical Memory State bits Input bits Input qubits Qubits after quantum operations Classical computer Quantumcomputer 1.Let us observe that the memory is classical 2.It can be realized with D, T, JK flip-flops, or other classical circuits.

12. QSMs with QUANTUM Memory (QSM-CM)

13. Model II: QSMs with Quantum Memory (QSM-QM). Model II: QSMs with Quantum Memory (QSM-QM). External quantum memory required. Re-initialization required only when an external input changes. Measurement performed only when computation ceases. Quantum Array Initialization Measurement and read out. Quantum Memory State qubits Input bits Input qubits Output bits Qubits after quantum operations 1.This model uses a quantum memory. 2.Feedback is realized by quantum memory. 3.There are several models of quantum memory as a special separate unit

14. Quantum State Machines With State Retention (QSM-SR)

15. Model III: Quantum State Machines With State Retention (QSM-SR) Model III: Quantum State Machines With State Retention (QSM-SR) Qubits retain their state. No external feedback required. Quantum Array Initialization Measurement and read out. Input bits Input qubits Qubits after quantum operations Output bits 1.This circuit uses internal memory in each qubit, the same as in normal quantum combinational circuit discussed so far in the class. 2.Special way of initialization and measurement is needed. 3.This will be discussed in next lectures.

16. 3.2 Models of Classical Controller of QFSM The Controller issues pulses for: 1.Initialization of qubits. 2.Generating the quantum array 3.Measurement operation. Only this part is quantum The controller is a classical circuit (computer)

17. Simple Model of a Controller The controller is modeled as an FSM with three states: I – I (Initialize), – C – C (Calculate), M – M (Measure). Ti Tm The controller continuously samples two inputs Ti and Tm. Ti Trigger generator monitors changes in external inputs and sets Ti. TiI – When Ti is set, controller jumps to I state. Ti C – After initialization, controller resets Ti and jumps to C state. TmM When Tm is set, controller jumps to M state and issues pulses for measurement.

18. 18 The transformations of blocks of quantum gates to the pulses level. The transformations of blocks of quantum gates to the pulses level.

19. Variants of the controller Two variants of the controller based on its behavior during the ‘C’ state: – Controller with Repeated Quantum Array (C- RQA). – Controller with Single Quantum array (C-SQA)

20. 3.3 High Level Synthesis 1.Synchronous 1.Very similar to classical FSM design. 2.Derive a truth table from the abstract specification (STG, SG etc) and pass it to DCARL. 2.Asynchronous 1.Muller Method 1.Uses generalized Muller C-elements (gC). 2.High level synthesis involves developing the functions for the Set and Reset inputs of the gC element. 2.Huffman Method 1.A state encoding which is free from critical races is chosen. 2.High level synthesis involves developing a next state table There are two types of state machines: synchronous and asynchronous

21. 3.4 Low Level Synthesis DCARL – software synthesis tool for Quantum Permutative Circuits DCARL – software synthesis tool for Quantum Permutative Circuits – Existing synthesis techniques like the MMD algorithm handle only completely specified reversible functions. – Output and Next state functions of state machines have the possibility of large number of “don’t cares” – DCARL allows incompletely specified functions to be handled by the MMD synthesis package. It ensures that there is a “one to one” correspondence between the inputs and outputs of a function. It converts irreversible functions to their reversible equivalents by adding ancilla bits. – Eg: Completely specified reversible function InputOutput 0010 01 1000 11 This is a standard truth table of a reversible function

22. Example: Combinational gate – Feynman – as a simple quantum state machine Eg: Irreversible function XOR gate Incompletely specified function Gate level realization Fully specified reversible function InputOutput 00X0 01X1 10X1 11X0 Input (S 1 S o ) Output (S 1 S o ) 00 01 1011 10 InputOutput 000 011 101 110 DCARL Step 2: MMD DCARL Step 1:

23. Example 1: Design of Synchronous quantum state machine using DCARL Specification : Step 1: Step 1: Transform to truth table

24. Step 2: Run DCARL Output of DCARL: Step 3: Implement complete QSM and controller 0000 0101 1010 1111 0000 0101 0101 Example 1 (cont): Design of Synchronous quantum state machine using DCARL

25. Example 2: Asynchronous design: Using Huffman method. Step1 Step1: Convert to next state table. Asynchronous toggle circuit: 0010 0001 11 01 1110 ab t 0 1 00 01 11 10 t+ = rising edge of a signal t- = falling edge of a signal

26. Step 2: Step 2: Derive next state functions 01 00 1 0 11 ab t 0 1 00 01 11 10 a = at b t + 00 01 1 1 10 ab t 0 1 00 01 11 10 b = a t b t + Note: DCARL can be used in this step Now these equations will be realized in one of QFSM models

27. Step 3: Implement the QSM QSM-SR a) Realization as a QSM-SR a = at b t + b = a t b t + + a = at b t + a+ b+ inputs New states garbages a+ b+ ( State Retention In place ( State Retention )memory

28. Explanation of feedback circuit a b b 0 0 a a + = ab a b b 0 0 a ab Above circuit is equivalent to the below circuit 0 a + = ab ab= garbage

29. b) Realization as QSM-QM Step 3: Implement the QSM a = at b t + b = a t b t + New states

30. c) Realization as a QSM-CM Step 3: Implement the QSM The same circuit as in last slide

31. Generalized Muller C-elements (gC).

32. Step 2: Derive set and reset functions for the output signals. Step 3: Implement the QSM using gC elements. gC element QSM-SR a) Implementation as a QSM-SR Example 3 (cont): Asynchronous Design: Using Muller Method. See next slides for details of calculations

33. Example 3: Asynchronous Design: Using Muller Method. Asynchronous toggle circuit: Step 1: Encode state graph with “R” (excited to rise) and “F” (excited to fall) signals.

34. Example 3: Asynchronous Design: Using Muller Method. Calculating excitation inputs set (a) and reset (a) for C-element a

35. Example 3: Asynchronous Design: Using Muller Method. Calculating excitation inputs set (a) and reset (a) for C-element b

36. Step 2: Derive set and reset functions for the output signals. Step 3: Implement the QSM using gC elements. gC element QSM-SR a) Implementation as a QSM-SR Example 3 (cont): Asynchronous Design: Using Muller Method.

37. QSM-QM b) Implementation as a QSM-QM QSM-CM c) Implementation as a QSM-CM Example 3 (cont): Asynchronous Design: Using Muller Method.

38. QSM-QM b) Implementation as a QSM-QM Example 3 (cont): Asynchronous Design: Using Muller Method.

39. QSM-CM c) Implementation as a QSM-CM Example 3 (cont): Asynchronous Design: Using Muller Method.

40. 5. Conclusions 1.Practical models for QSMs and their classical controllers have been developed. 2.A synthesis flow which allows us to design QSMs from abstract specifications has been developed. 3.Synchronous as well as asynchronous techniques for high level synthesis of QSMs have been demonstrated. 4.DCARL software for automating low level synthesis (i.e. quantum array level) has been demonstrated.