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1. A Computer Architecture For Quantum Programming (Using QC modules in a classical computer from C++) From: http://arxiv.org/PS_cache/cs/pdf/0103/0103009.pdf By Tom Bersano For EECS598 Quantum Computing Thursday, December 13, 2001

2. The Goal: Quantum Programming • Classical Computer Using QC Module • Why? • A hybrid classical/quantum system can provide advantages of both (ease of programmability/interface AND quantum speedup advantages). • Example: • Classical computer provides nice interface (data transfer, etc.) and uses convenient compilers, etc. • Part of algorithm that would best be solved in a QC are identified, a Quantum Circuit is compiled and implemented, and used as a ‘black-box function call’.

3. Example: Grover’s Algorithm:Find x in N … Exponential speedup over classical Qbitset run_Grover(bool(*f)(int), int n) { int repetitions = sqrt(pow(2.0,n)); Qop phase_oracle(f,n); Qop invert_zero(f_0,n); Qop mixer = QHadamard(n); Qop invert_mean =mixer & invert_zero & mixer; Qop grover_step = phase_oracle & invert_mean; Qreg input(n); mixer(input); for (int i=0; i<repetitions; ++i) { grover_step(input); } return input.measure(); } Define Quantum Operations Define Q-Register Invoke Q-Operation On Q-Reg

4. Often optimization of Q-circuit depends on code and equivalent circuit structure alone, and not on register values Solution: Generate entire circuit description as data structure, and apply algebraic simplifications before ever allocating quantum registers Optimizations Qreg myreg(size); for (int i=0; i<(size-1); i++) Hadamard_2(myreg, i); size = 6 Qop circuit; for (int i=0; i<(size-1); i++) circuit << QHadamard(2).offset(i); Qreg myreg(size); circuit(myreg);

5. C++ code Hardwareindependent assembly/machine code Hardware Dependent # registers, primitive operations, Speed Limitations (#cycles per instr.) Etc. Program vs. Architecture:A Classical Computer Example COMPILEDINTO

6. Program vs. Architecture in QC:The QRAM Architecture • Quantum Random Access Machine Classical System void main() { Obj n1; … for(…) { … } print(result); } Part of algorithm that is exponential in classical systems

7. Program vs. Architecture in QC:The QRAM Architecture • Quantum Random Access Machine Classical System Black Box QRAM: Memory/ Registers (Q-Bits) void main() { Obj n1; … … QC_call … print(result); } Encoder/ Decoder Interface to QRAM ENCODES PROBLEM CALLS RETURNS READS ANSWER

8. Quantum Programming Classical System Black Box QRAM: Memory/ Registers (Q-Bits) Quantum Circuit Description: void main() { Obj n1; … … QC_call … print(result); } Encoder/ Decoder Interface to QRAM ENCODES PROBLEM CALLS RETURNS READS ANSWER Compiler Compiled Code for Classical Computer

9. Quantum Programming Classical System Black Box QRAM: Memory/ Registers (Q-Bits) Quantum Circuit Description: void main() { QReg q(5); … … QC_call … print(result); } Encoder/ Decoder Interface to QRAM ENCODES PROBLEM CALLS RETURNS READS ANSWER Compiler Compiled Code for Classical Computer

10. Desirable Properties • Completeness • Language maps to and from every possible Q. Circuit • Separability • QC in algorithm are always separate modules. • Classical Extension / Expressivity • Example: C++ with QC function calls • Hardware independence • C++ works on old 386, Athlon, MAC, etc. • QC extensions must work regardless of physical impl.

11. Q.Registers: Declaration: Qreg a_register(5);//allocates a register with 5 qubits int the size = a_register.size(); //gets size of the register Addressing: Qreg a_qubit = a_register; //selects the 4th qubit from a register Qreg a_subreg = a_register(2,5); //selects 5 qubits starting at the 3rd one Concatenation: Qreg new reg = a_subreg & a_qubit; //concatenates the two registers Resizing: a_register += 5; //adds 5 qubits to my register a_register -= 3; //drops 3 qubits from my register Quantum Programming Data Structures:

12. Quantum Memory: The Qreg Object

13. Quantum Register Management from Classical Side: Q-Bit Swapping • Another Advantage of Hybrid System: • Swapping bits in Quantum Registers can be difficult and unnecessary… just use classical computation to keep track of bit permutations for the final solution.

14. Quantum Operator Primitives • High Level Primitives (HLP) are implemented out of any chosen set of Low Level Primitives (LLP) at ‘compile/run’ time. List ctrls = (0,4,5); List targets = (1,2,6); Qop my op = QCnot(ctrls, targets); • Example of complete set of Quantum Low Level Primitives:

15. Computational Primitives

16. Q.Operators: (list not complete) Application: an_operator(a_register); runs the circuit onto the register Fixed arity quantum operators: Qop my_op = QHadamard(7); Hadamard gates acting on First 7 qubits Qop my_op = QCnot(ctrls, targets); Controlled operators: Qop a_controlled_op(U, 5); creates a U conditioned by 5 qubits Operators for classical functions: Qop an_oracle = Qop(f,3,5); oracle for f with n = 3 and m = 5 Qop a_phase_oracle = Qop(g,4); phase oracle for g with n = 4 (m is 1) Operators Composition: Qop composed = part_1 & part_2; composes two Qops into a third Qop my_operator &= an_operator; extends my operator with an operator my_operator << an_operator; moves an operator into my operator Quantum Programming Data Structures:

17. Quantum Operators:The Qop object

18. Addition using QFt and composition F N4 F-1 N4+M4

19. Addition using QFt and composition Qop build_three_adder(int size){ Qop phase_shifts; for (int i=0; i<size; ++i) phase_shifts << QCondPhase(size-i, i+1).offset(i); Qop transform = (QFourier(size) & QSwap(size)).offset(size); Qop adder_2 = transform & phase_shifts & (! transform); Qop adder_3 = (adder_2 >> size); adder_3 << adder_2.split(size, size); return adder_3; }

20. Creating Oracles/Pseudo-Classical Operators: • Any classical non-reversible function has an equivalent reversible function • Any reversible function can be converted efficiently into a quantum mechanical function • Thus, in theory, we can build a quantum mechanical oracle on a q-register for any given classical oracle function, but we must use it appropriately to get any speedup advantage… • My opinion: no straightforward way (yet) to build Quantum Oracles that implement classical functions (without being explicitly ‘called’ for each input combination and thus losing speedup adv.)

21. Assumptions about Q.Hardware • Classical Case • Example: if ALU doesn’t have multiplication or permutation, their complexity is linear in #bits • Bus size, cache size, etc. matter • Single and 2-QBit primitive gates can be executed in constant time • For non-adjecent QBits time complexity may scale linearly by distance (e.g., number of swaps required) so complexity is worse than algorithmic one. • Parallel quantum execution assumed but only exploited in homogeneous gates

22. Conclusions – part 1 • C++ Library for Q.Programming already implemented • http://meitner.tn.infn.it/~bettelli/qcomp/qlang/index.pl • QC hardware not ready, can work with simulator for the moment. • Accomplishment: • Allows rewriting algorithms in formal language • Provides a general Framework for: • Comparing/Testing different implementations/algorithms for: • high level simplification and optimization routines for quantum circuits • schemes for high level to low level and hardware independent to dependent translation routines for quantum circuits • hardware architectures for the execution of quantum code (with timing simulations); • robustness of error correction codes and fault tolerant quantum computation with respect to generic error models • Algorithm design: • having an high level interface for the specification of algorithms which are to be fed into quantum simulators; • quantum programming(when quantum computers will be ready)

23. Conclusions – part 2 • Weaknesses • Says that quantum circuit optimizations can be done but doesn’t tell you how to do them • Says that pseudo-classical oracles can in theory be created but doesn’t tell you how to do so