SQIR & VOQC

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SQIR is a Small Quantum Intermediate Representation for quantum programs. Its intended use is as an intermediate representation in a Verified Optimizer for Quantum Circuits (VOQC), but we have also used it to implement verified versions of several quantum algorithms.

We first presented SQIR and its use in VOQC in our paper A Verified Optimizer for Quantum Circuits at POPL 2021. We provide additional details of verifying SQIR programs (including QPE and Grover's) in our paper Proving Quantum Programs Correct, presented at ITP 2021. We describe a SQIR formalization of Shor's factoring algorithm in our draft A Formally Certified End-to-End Implementation of Shor's Factorization Algorithm.

This repository contains our Coq formalization of SQIR and VOQC as well as several verified quantum algorithms. If you are interested in running the VOQC compiler, then you should look at our OCaml library (inQWIRE/mlvoqc) or Python library (inQWIRE/pyvoqc) instead. The OCaml library is extracted from our Coq definitions and the Python library is a wrapper around the OCaml library.

If you are interested in learning more about formal verification of quantum programs in general, we recommend Robert Rand's Verified Quantum Computing tutorial.

Table of Contents

• SQIR & VOQC
  • Table of Contents
  • Setup
  • Compilation
  • Using With Other Projects
  • Directory Contents
    • SQIR
    • VOQC
    • examples
  • Acknowledgements
  • Citations

Setup

To compile SQIR and VOQC, you will need Coq and QuantumLib (version 1.7.0). In order to build the Shor proof, you will also need the Coq Interval package and the coq-euler library. We strongly recommend using opam to install Coq and opam switch to manage Coq versions. We currently support Coq versions 8.16-8.20. If you run into errors when compiling our proofs, first check your version of Coq (coqc -v).

Assuming you have opam installed (following the instructions in the link above), follow the steps below to set up your environment.

# environment setup
opam init
eval $(opam env)

# install some version of the OCaml compiler in a switch named "voqc"
opam switch create voqc 4.13.1
eval $(opam env)

# install Coq -- this will take a while!
opam install coq

# install the QuantumLib library
opam repo add coq-released https://coq.inria.fr/opam/released
opam update
opam install coq-quantumlib.1.7.0

# Optional, if you want to compile the proofs in examples/shor
opam install coq-interval
opam pin coq-euler https://github.com/taorunz/euler.git

Notes:

• Depending on your system, you may need to replace 4.13.1 in the instructions above with something like "ocaml-base-compiler.4.13.1". Any recent version of OCaml should be fine.
• We require Coq version >= 8.16.
• opam error messages and warnings are typically informative, so if you run into trouble then make sure you read the console output.

Compilation

After following the setup directions above, you can run make to compile the core files of SQIR, make voqc to compile proofs about VOQC, make examples to compile proofs of correctness for quantum algorithms (excluding those in examples/shor), and make shor to compile proofs about Shor's algorithm. Use make all to compile everything.

Our proofs are resource intensive so expect make all to take a little while. If you have cores to spare, then you can speed things up by compiling with the -j flag (e.g., make all -j8). On a 2015 dual-core MacBook Pro running Coq version 8.15.2, compilation takes around 30 minutes.

Using With Other Projects

To install SQIR, run opam pin coq-sqir https://github.com/inQWIRE/SQIR.git

To pull subsequent updates, run opam install coq-sqir.

To import SQIR files, use Require Import SQIR.FILENAME

Directory Contents

SQIR

Definition of the SQIR language.

• DensitySem.v : Density matrix semantics for general SQIR programs.
• Equivalences.v : Verified circuit equivalences for peephole optimizations.
• ExtractionGateSet.v : Expanded gate set used for extraction.
• GateDecompositions.v : Verified optimized decompositions for CH, CU1, CU2, CU3, CCU1, CSWAP, C3X, and C4X.
• NDSem.v : Non-deterministic semantics for general SQIR programs.
• DiscreteProb.v : Utilities to describe running a quantum program and sampling from the output probability distribution.
• SQIR.v : Definition of the SQIR language.
• UnitaryOps.v : Utilities for manipulating unitary SQIR programs.
• UnitarySem.v : Semantics for unitary SQIR programs.

VOQC

Verified transformations of SQIR programs. The optimizations and mapping routines extracted to our OCaml library (inQWIRE/mlvoqc) are all listed in Main.v. So this file is a good starting point for getting familiar with VOQC.

The rest of the files in the VOQC directory can be split into the following categories:

• Utilities

  • GateSet.v : Coq module for describing a quantum gate set.
  • IBMGateSet.v : "IBM" gate set {U1, U2, U3, CX}.
  • NonUnitaryListRepresentation.v : List representation of non-unitary SQIR programs.
  • RzQGateSet.v : "RzQ" gate set {H, X, Rzq, CX}.
  • FullGateSet.v : Full gate set {I, X, Y, Z, H, S, T, Sdg, Tdg, Rx, Ry, Rz, Rzq, U1, U2, U3, CX, CZ, SWAP, CCX, CCZ}.
  • UnitaryListRepresentation.v : List representation of unitary SQIR programs; includes utilities for manipulating program lists and gate set-independent proofs.

• Optimizations over unitary programs, inspired by those in Qiskit and Nam et al. [2018]

  • ChangeRotationBasis.v : Auxiliary proof for Optimize1qGates.
  • GateCancellation.v : "Single-qubit gate cancellation" and "two-qubit gate cancellation" from Nam et al.
  • HadamardReduction.v : "Hadamard reduction" from Nam et al.
  • NotPropagation.v : "Not propagation" from Nam et al.
  • Optimize1qGates.v : Optimize1qGates from Qiskit.
  • RotationMerging.v : "Rotation merging using phase polynomials" from Nam et al.

• Optimizations over non-unitary programs

  • PropagateClassical.v : Track classical states to remove redundant measurements and CNOT operations.
  • RemoveZRotationBeforeMeasure.v : Remove single-qubit z-axis rotations before measurement.

• Mapping routines

  • ConnectivityGraph.v : Utilities for describing an architecture connectivity graph. Includes graphs for linear nearest neighbor and 2D grid architectures.
  • GreedyLayout.v : Generate a layout based on the input program and architecture.
  • Layouts.v : Utilities for describing a physical <-> logical qubit mapping (i.e., layout).
  • MappingConstraints.v : Utilities for describing a program that satisfies architecture constraints.
  • MappingGateSet.v : Mapping gate set U + {CX, SWAP}, where U is an arbitrary set of single-qubit gates.
  • MappingValidation.v : Check whether two programs (which differ only in SWAP placement) are equivalent.
  • SwapRoute.v: Simple mapping for an architecture described by a directed graph.

• Experimental extensions

  • BooleanCompilation.v : Compilation from boolean expressions to unitary SQIR programs.

examples

Examples of verifying correctness properties of quantum algorithms.

• Deutsch.v : Deutsch algorithm
• DeutschJozsa.v : Deutsch-Jozsa algorithm
• ghz/ : GHZ state preparation
• Grover.v : Grover's algorithm
• QPE.v : Simplified quantum phase estimation
• shor/ : Shor's algorithm, including general quantum phase estimation (use make shor to compile separately, see the README in the shor directory for more details)
• Simon.v : Simon's algorithm
• Superdense.v : Superdense coding
• Teleport.v : Quantum teleportation
• Utilities.v : Miscellaneous utilities used in multiple examples
• Wiesner.v : Wiesner's quantum money, contributed by Adrian Lehmann

Acknowledgements

This project is the result of the efforts of many people. The primary contacts for this project are Kesha Hietala (@khieta) and Robert Rand (@rnrand). Other contributors include:

• Le Chang
• Akshaj Gaur
• Aaron Green
• Kesha Hietala
• Shih-Han Hung
• Adrian Lehmann
• Liyi Li
• Yuxiang Peng
• Robert Rand
• Kartik Singhal
• Runzhou Tao
• Finn Voichick

This project is supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Quantum Testbed Pathfinder Program under Award Number DE-SC0019040 and the Air Force Office of Scientific Research under Grant Number FA95502110051.

Citations

If you use SQIR or VOQC in your work, please cite our papers.

@article{hietala2021verified,
  title         = {A Verified Optimizer for {{Quantum}} Circuits},
  author        = {Hietala, Kesha and Rand, Robert and Hung, Shih-Han and Wu, Xiaodi and Hicks, Michael},
  year          = {2021},
  month         = jan,
  journal       = {Proceedings of the ACM on Programming Languages},
  volume        = {5},
  number        = {POPL},
  eid           = {37},
  pages         = {37},
  numpages      = {29},
  doi           = {10.1145/3434318},
  archiveprefix = {arXiv},
  eprint        = {1912.02250},
  url           = {https://github.com/inQWIRE/SQIR}