Schrödinger and Schrödinger-Feynman simulators for quantum circuits.

## Project description

# qsim and qsimh

Quantum circuit simulators qsim and qsimh. These simulators were used for cross entropy benchmarking in [1].

[1], F. Arute et al, "Quantum Supremacy Using a Programmable Superconducting Processor", Nature 574, 505, (2019).

## qsim

qsim is a Schrödinger full state-vector simulator. It computes all the *2 ^{n}*
amplitudes of the state vector, where

*n*is the number of qubits. Essentially, the simulator performs matrix-vector multiplications repeatedly. One matrix-vector multiplication corresponds to applying one gate. The total runtime is proportional to

*g2*, where

^{n}*g*is the number of 2-qubit gates. To speed up the simulator, we use gate fusion [2] [3], single precision arithmetic, AVX/FMA instructions for vectorization and OpenMP for multi-threading.

[2] M. Smelyanskiy, N. P. Sawaya, A. Aspuru-Guzik, "qHiPSTER: The Quantum High Performance Software Testing Environment", arXiv:1601.07195 (2016).

[3] T. Häner, D. S. Steiger, "0.5 Petabyte Simulation of a 45-Qubit Quantum Circuit", arXiv:1704.01127 (2017).

## qsimh

qsimh is a hybrid Schrödinger-Feynman simulator
[4]. The lattice is split into two parts
and the Schmidt decomposition is used to decompose 2-qubit gates on the
cut. If the Schmidt rank of each gate is *m* and the number of gates on
the cut is *k* then there are *m ^{k}* paths. To simulate a circuit with
fidelity one, one needs to simulate all the

*m*paths and sum the results. The total runtime is proportional to

^{k}*(2*, where

^{n1}+ 2^{n2})m^{k}*n*and

_{1}*n*are the qubit numbers in the first and second parts. Path simulations are independent of each other and can be trivially parallelized to run on supercomputers or in data centers. Note that one can run simulations with fidelity

_{2}*F < 1*just by summing over a fraction

*F*of all the paths.

A two level checkpointing scheme is used to improve performance. Say, there
are *k* gates on the cut. We split those into three parts: *p+r+s=k*, where
*p* is the number of "prefix" gates, *r* is the number of "root" gates and
*s* is the number of "suffix" gates. The first checkpoint is executed after
applying all the gates up to and including the prefix gates and the second
checkpoint is executed after applying all the gates up to and including the
root gates. The full summation over all the paths for the root and suffix gates
is performed.

If *p>0* then one such simulation gives *F ≈ m ^{-p}* (for all the
prefix gates having the same Schmidt rank

*m*). One needs to run

*m*simulations with different prefix paths and sum the results to get

^{p}*F = 1*.

[4] I. L. Markov, A. Fatima, S. V. Isakov, S. Boixo, "Quantum Supremacy Is Both Closer and Farther than It Appears", arXiv:1807.10749 (2018).

## C++ Usage

The code is basically designed as a library. The user can modify sample aplications in apps to meet their own needs. The usage of sample applications is described in docs.

### Input format

Circuit input format is described in docs.

### Sample Circuits

A number of sample circuits are provided in circuits.

### Unit tests

Unit tests are located in tests. The Google test framework is used. To build and run all tests, navigate to the test directory and run:

```
make run-all
```

This will compile all test binaries to files with `.x`

extensions, and run each
test in series. Testing will stop early if a test fails.

To clean up generated test files, run `make clean`

from the test directory.

## Cirq Usage

Cirq is a framework for modeling and invoking Noisy Intermediate Scale Quantum (NISQ) circuits.

To run qsim on Google Cirq circuits, or just to call the simulator from Python, see docs.

## Disclaimer

This is not an officially supported Google product.

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