engineering models and design methods for quantum state machines

Engineering Models

and Design Methods

for Quantum State Machines.

• 1. Synthesis Flow for Quantum State Machines (QSM)– Models of QSM

– Models of Controller

– High Level Synthesis

– Low Level Synthesis• DCARL

• 2. Examples

1.2 What is this lecture about?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.

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.

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.

Quantum Gate operates in time Quantum Gate operates in time not in spacenot 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. Cnext=ab c

Note: When c=1, Nand operation is performed on ab!!

Illustration:

C

a

b

c

1. Explain in detail how quantum gate operates in time, based on controlled pulses.

2. Classical computer creates quantum pulses, such as EM

Quantum Computing requires classical Quantum Computing requires classical computer for operationcomputer 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).

INITIALIZATIONINITIALIZATION

COMPUTATIONCOMPUTATION

MEASUREMENTMEASUREMENT

timeCLASSICAL CLASSICAL

COMPUTER COMPUTER WORKS AS A WORKS AS A CONTROLERCONTROLER

3. Synthesis Flow of designing quantum state 3. Synthesis Flow of designing quantum state machinesmachines

3.1 Models of QSM3.1 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.

QSMs with QSMs with Classical Classical Memory Memory

(QSM-CM)(QSM-CM)

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

State bits

Input bits

Input qubits

Qubits after quantum operations

Classical Classical computercomputer

QuantumQuantumcomputercomputer

1.1. Let us observe that the memory is classicalLet us observe that the memory is classical2.2. It can be realized with D, T, JK flip-flops, or other classical It can be realized with D, T, JK flip-flops, or other classical

circuits.circuits.

QSMs with QSMs with QUANTUMQUANTUM

Memory Memory

(QSM-CM)(QSM-CM)

Model IIModel II: QSMs with Quantum Memory (QSM-QM).: 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 ArrayInitializationMeasurement and read out.

Quantum Memory

State qubits

Input bits

Input qubits

Output bitsQubits

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

Quantum State Quantum State Machines Machines

With State Retention With State Retention

(QSM-SR)(QSM-SR)

Model IIIModel III: Quantum State Machines With State Retention (QSM-SR): Quantum State Machines With State Retention (QSM-SR)Qubits retain their state.No external feedback required.

Quantum ArrayInitializationMeasurement 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.

3.2 Models of Classical 3.2 Models of Classical Controller of QFSMController 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)

Simple Model of a ControllerSimple Model of a Controller• The controller is modeled as an FSM with three states:

– II (Initialize),

– CC (Calculate),

– M M (Measure).

• The controller continuously samples two inputs Ti Ti and TmTm.

• Trigger generator monitors changes in external inputs and sets TiTi.– When TiTi is set, controller jumps to II state.

– After initialization, controller resets Ti Ti and jumps to CC state.

• When TmTm is set, controller jumps to MM state and issues pulses for measurement.

18

• The The transformations transformations

of blocks of of blocks of quantum quantum gates to the gates to the pulses level.pulses level.

Variants of Variants of the controllerthe 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)

3.3 High Level Synthesis3.3 High Level Synthesis

1.1. SynchronousSynchronous1. Very similar to classical FSM design.

2. Derive a truth table from the abstract specification (STG, SG etc) and pass it to DCARL.

2.2. AsynchronousAsynchronous

1.1. Muller Method 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.

2.2. Huffman MethodHuffman Method1. 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

3.4 Low Level Synthesis3.4 Low Level Synthesis• DCARL – software synthesis tool for Quantum DCARL – software synthesis tool for Quantum

Permutative CircuitsPermutative 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

Input Output

00 10

01 01

10 00

11 11

This is a standard truth table of a reversible function

Example: Combinational gate – Feynman Example: Combinational gate – Feynman – as a simple quantum state machine– as a simple quantum state machine

• Eg: Irreversible function

• XOR gate

• Incompletely specified function

• Gate level realization

• Fully specified reversible function

Input Output

00 X0

01 X1

10 X1

11 X0

Input(S1So)

Output (S1So)

00 00

01 01

10 11

11 10

Input Output

00 0

01 1

10 1

11 0

DCARLStep 2:

MMDDCARLStep 1:

Example 1: Design of Example 1: Design of Synchronous Synchronous quantum state machine using DCARLquantum state machine using DCARL

Specification:Step 1: Step 1: Transform to truth table

Step 2: Run DCARL

Output of DCARL:

Step 3: Implement complete QSM and controller

0

0

0

1

1

0

1

1

0

0

0

1

0

1

Example 1 (cont): Design of Example 1 (cont): Design of SynchronousSynchronous quantum state machine using DCARLquantum state machine using DCARL

Example 2: Asynchronous design: Example 2: Asynchronous design: Using Huffman method. Using Huffman method.

• Step1Step1: Convert to next state table.Asynchronous toggle circuit:

00 10

00 01

11 01

11 10

ab

t0 1

00

01

11

10t+ = rising edge of a signal

t- = falling edge of a signal

Step 2: Step 2: Derive next state functions

0 1

0 0

1 0

1 1

ab

t0 1

00

01

11

10

a = at b t+

0 0

0 1

1 1

1 0

ab

t0 1

00

01

11

10

b = a t b t+

Note: DCARL can be used in this stepNow these equations will be realized in one of QFSM models

Step 3: Step 3: Implement the QSMImplement the QSM• a) Realization as a QSM-SRQSM-SR

a = at b t+

b = a t b t+

b = a t b t+ a = at b t+

a+

b+

inputs

New states

garbages

a+

b+

In place ( State ( State RetentionRetention )memory

Explanation of feedback circuitExplanation of feedback circuitab b0

0

a

a+ = ab

ab b0

0

a

ab

Above circuit is equivalent to the below circuitAbove circuit is equivalent to the below circuit

0

a+ = ab

ab= garbage

• b) Realization as QSM-QM

Step 3: Step 3: Implement the QSMImplement the QSM

a = at b t+

b = a t b t+New states

c) Realization as a QSM-CM

Step 3: Step 3: Implement the QSMImplement the QSM

The same circuit as in last slide

Generalized Muller C-elements (gC).

Step 2: Derive set and reset functions for the output signals.

Step 3: Implement the QSM using gC elements.

gC element

gC element

a) Implementation as a QSM-SRQSM-SR

Example 3 (cont): Asynchronous Design: Example 3 (cont): Asynchronous Design: Using Muller Method.Using Muller Method.

See next slides for details of calculations

Example 3: Asynchronous Design: Using Example 3: Asynchronous Design: Using Muller Method.Muller Method.

Asynchronous toggle circuit: Step 1: Encode state graph with “R” (excited to rise) and “F” (excited to fall) signals.

Example 3: Asynchronous Example 3: Asynchronous Design: Using Muller Method.Design: Using Muller Method.

Calculating excitation inputs set (a) and reset (a) for C-element a

Example 3: Asynchronous Example 3: Asynchronous Design: Using Muller Method.Design: Using Muller Method.

Calculating excitation inputs set (a) and reset (a) for C-element b

Step 2: Derive set and reset functions for the output signals.

Step 3: Implement the QSM using gC elements.

gC element

gC element

a) Implementation as a QSM-SRQSM-SR

Example 3 (cont): Asynchronous Design: Example 3 (cont): Asynchronous Design: Using Muller Method.Using Muller Method.

b) Implementation as a QSM-QMQSM-QM c) Implementation as a QSM-CMQSM-CM

Example 3 (cont): Asynchronous Design: Example 3 (cont): Asynchronous Design: Using Muller Method.Using Muller Method.

b) Implementation as a QSM-QMQSM-QM

Example 3 (cont): Asynchronous Design: Using Muller Method.Example 3 (cont): Asynchronous Design: Using Muller Method.

c) Implementation as a QSM-CMQSM-CM

Example 3 (cont): Asynchronous Design: Using Muller Method.Example 3 (cont): Asynchronous Design: Using Muller Method.

5. Conclusions5. Conclusions1. 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.