RAndom Circuit Block Encoded Matrix (RACBEM)

This is a python module implementing RACBEM using IBM's Qiskit. It can used in various quantum linear algebra problems.

Dependencies and Installing Instruction

• Pyhton3.7, IBM Qiskit, QuTiP, numpy, scipy

• Installing Instruction:

  1. Install Anaconda for python-package management, avaliable at https://www.anaconda.com/products/individual

     Go to the terminal after installing Anaconda.

  2. If you have already had a Python3.7 environment, just skip this step.

     conda create --name=$ENV_NAME python=3.7 # have the environment use Python 3.7

     source activate $ENV_NAME

  3. Make sure the modules for scientific computing, numpy, scipy, are installed.

     conda install numpy scipy

     You are also suggested to install matplotlib for visualization and Cython due to the requirement of QuTiP.

     conda install matplotlib cython

  4. Install the package pickle to load/save data locally.

     conda install pickle

  5. Install QuTiP. If the installation is terminated due to missing dependent packages, please check the document provided by QuTiP, http://qutip.org/docs/latest/installation.html

     conda install qutip

     If conda doesn't work, try pip for instead.

     pip install qutip

     You can also install directly from source if both conda and pip don't work but it's rare. See Installing from Source in the document for details. http://qutip.org/docs/latest/installation.html

  6. Install IBM Qiskit. See the document for details. https://qiskit.org/documentation/install.html

     pip install qiskit

  7. In your python environment, test whether you successfully install IBM Qiskit.

     python

     >>> import qiskit

     If the module is loaded, the installation is all set.

  8. Run our demo code.

     python main_test.py

You are suggested to get and save locally an IBM account which is used frequently when using IBM Qiskit. See Access IBM Quantum Systems in the document for details. https://qiskit.org/documentation/install.html

Introduction

This is an implementation of the a RAndom Circuit Block Encoded Matrix (RACBEM) and its Hermitian conjugate. It is then used to build a quantum singular value circuit using the method of quantum singular value transformation (QSVT).

Take a RAndom Circuit Block Encoded Matrix (RACBEM), this function uses a quantum signal processing circuit to evaluate the matrix inverse, using the method of quantum singular value transformation (QSVT). This implements a (non-Hermitian) block-encoding of a Hermitian matrix manually.

References

• Yulong Dong, Lin Lin. Random circuit block-encoded matrix and a proposal of quantum LINPACK benchmark. arXiv: 2006.04010
• Y. Dong, X. Meng, K. B. Whaley, and L. Lin. Efficient Phase Factor Evaluation in Quantum Signal Processing. arXiv: 2002.11649
• A. Gilyén, Y. Su, G. H. Low, and N. Wiebe. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 193–204, 2019

Citing our work

If you find our work useful or you use our work in your own project, please consider to cite our work.

The authors

We hope that the package is useful for your application. If you have any bug reports or comments, please feel free to email one of the software authors:

• Yulong Dong, [email protected]

• Lin Lin, [email protected]

Notice

It's possible to get the following warning message when importing racbem. It's due to a problem of QuTiP (github.com/qutip/qutip/issues/1205). Please just ignore it because it doesn't affect the computation.

It may happen in MacOS, Windows or Linux.

/Users/$USER/.pyxbld/temp.macosx-10.9-x86_64-3.8/pyrex/qutip/cy/openmp/parfuncs.cpp:614:10: fatal error: 'src/zspmv_openmp.hpp' file not found
#include "src/zspmv_openmp.hpp"
         ^~~~~~~~~~~~~~~~~~~~~~
1 error generated.

Running the demo

python main_test.py

The output should be like the follows.

retrieve architecture from IBM Q and save locally at: ibmq_burlington_backend_config.pkl

kappa=5, sigma=1.00, polynomial approximation error=1.902e-02

Generic RACBEM
singular value (A) = 
 [0.99  0.979 0.952 0.79  0.694 0.306 0.248 0.204]
Job Status: job has successfully run
succ prob (exact)     =  0.22205160210017913
succ prob (noiseless) =  0.23183536309935282
succ prob (measure)   =  0.2242431640625

Hermitian RACBEM
singular value (A) = 
 [0.963 0.91  0.908 0.802 0.397 0.292 0.289 0.236]
condition number (A)  = 4.074
||A - A^\dagger||_2   = 2.675e-15

Generating phase factors using QSPPACK

To implement QSVT, a set of phase factors which characterizes the target function is needed. You can generate phase factors by using QSPPACK.

Installing Instruction:

1. Download the latest source code in our Github repository. https://github.com/qsppack/QSPPACK

2. Go to the directory where you save QSPPACK and run the following command in Matlab terminal.

   >> startup

Running the demo:

We provide a demo code to generate phase factors which solve QLSP whose condition number is upper bounded. You can follow the steps below after you set up QSPPACK.

1. Open Remez.ipynb in jupyter notebook. Make sure you installed Julia language. See https://julialang.org for details.

   Run jupyter notebook or LANG=zn jupyter notebook in terminal under the directory of RACBEM to open the notebook.

2. Run the code in the notebook. A data file coef_5_6.mat will apear in your directory. A polynomial approximation is saved in that file.

3. In Matlab terminal under the directory of RACBEM, run >> GeneratePhi(5,6). You will get 2 figures, the following messages, and a txt file phi_inv_5.txt in which the informations about phase factors are saved.

   approx error (inf) of coef = 0.0223098
   extra scaling factor = 1.17326
   total scaling factor = 5.86631
   L-BFGS solver started 
   iter          obj  stepsize des_ratio
      1  +3.3827e-03 +1.00e+00 +4.95e-01
      2  +1.0600e-03 +1.00e+00 +7.08e-01
      3  +8.8408e-05 +1.00e+00 +6.04e-01
      4  +3.9504e-06 +1.00e+00 +6.05e-01
      5  +6.4384e-08 +1.00e+00 +5.41e-01
      6  +1.0029e-09 +1.00e+00 +5.38e-01
      7  +2.7395e-11 +1.00e+00 +5.60e-01
      8  +5.8496e-14 +1.00e+00 +5.14e-01
      9  +7.4040e-17 +1.00e+00 +5.15e-01
     10  +8.5125e-21 +1.00e+00 +5.04e-01
   iter          obj  stepsize des_ratio
     11  +5.2673e-24 +1.00e+00 +5.06e-01
     12  +7.8591e-27 +1.00e+00 +5.16e-01
   Stop criteria satisfied.
   - Info: 		QSP phase factors --- solved by L-BFGS
   - Parity: 		even
   - Degree: 		6
   - Iteration times: 	12
   - CPU time: 	0.0 s
   approx error (inf) of polynomial = 1.384e-13
   approx error (inf) of g circ h = 1.902e-02