In the past three months, I participated my first GSoC (Google Summer of Code) and working on the Julia package ZXCalculus.jl. In this blog post, I will briefly introduce my work during GSoC 2020 in three parts, the first part is the high level interface of using ZXCalculus.jl as a quantum circuit simplification pass in YaoLang.jl. The second part is the lower level interface, and the thrid part is a benchmark with Python package PyZX.

## ZXCalculus.jl as a quantum circuit simplification pass

ZX-calculus is a graphical language for representing quantum states and operations. ZX-calculus is also used for simplifying quantum circuits. Let me show you how we can use ZXCalculus.jl to do circuit simplification.

Suppose that we have a quantum circuit with 24 gates as above. We can define this circuit with YaoLang.jl by using the macro @device easily. YaoLang.jl is a compiler for hybrid quantum-classical programs that are very practical in the current NISQ (noisy intermediate-scale quantum) era. Moreover, YaoLang.jl is integrated with ZXCalculus.jl. For more details about YaoLang.jl and quantum compilation, please read my second GSoC blog post.

 123456789101112131415161718192021222324252627282930 julia> using YaoLang; julia> @device optimizer = [:zx_teleport] function demo_circ_simp() 1 => shift($(7π/4)) 1 => H 1 => Rx($(π/4)) 4 => H @ctrl 1 4 => Z @ctrl 4 1 => X 1 => H 4 => H 1 => T 4 => shift(\$(3π/2)) 4 => X 1 => H 4 => S 4 => X 2 => S @ctrl 2 3 => X 2 => H @ctrl 2 3 => X 2 => T 3 => S 2 => H 3 => H 3 => S @ctrl 2 3 => X end demo_circ_simp (generic circuit with 1 methods) 

One can add an argument optimizer = [opts...] in the macro @device to simplify this circuit during compilation. Currently, there are only two optimization passes, :zx_clifford for Clifford simplification 1 and :zx_teleport for phase teleportation 2. For example, with optimizer = [:zx_teleport], the compiler will call the phase teleportation algorithm 2 in ZXCalculus.jl to simplify the circuit.

We can use the macro @code_yao to see the what circuit we have got. In this example, the gate number of the circuit has been decreased from 24 to 20.

 12345678910111213 julia> using YaoLang.Compiler julia> gate_count(demo_circ_simp) Dict{Any,Any} with 8 entries: "YaoLang.Rx(3.141592653589793)" => 2 "YaoLang.H" => 6 "YaoLang.Rx(0.7853981633974483)" => 1 "YaoLang.shift(4.71238898038469)" => 1 "YaoLang.shift(1.5707963267948966)" => 4 "YaoLang.shift(0.7853981633974483)" => 1 "@ctrl YaoLang.Z" => 1 "@ctrl YaoLang.X" => 4 

We can use YaoArrayRegister.jl to apply this simplified circuit on a quantum state.

 1234567891011 julia> using YaoArrayRegister; julia> circ_teleport = demo_circ_simp() demo_circ_simp (quantum circuit) julia> r = rand_state(4); julia> r |> circ_teleport ArrayReg{1, Complex{Float64}, Array...} active qubits: 4/4 

One can also load circuits from OpenQASM codes. OpenQASM is a quantum instruction. OpenQASM codes can be run on IBM Q devices. And quantum circuits can be stored as OpenQASM codes. I used the Julia package RBNF.jl (a Julia parser that parses code to restricted Backus-Naur form) to parse OpenQASM codes to ASTs, and then convert it to YaoIR, an intermediate representation for hybrid quantum-classical programs in YaoLang.jl. This makes it possible to read circuits from OpenQASM codes to ZXCalculus.jl via YaoLang.jl.

 12345678910 using YaoLang: YaoIR, is_pure_quantum using ZXCalculus lines = readlines("gf2^8_mult.qasm") src = prod([lines[1]; lines[3:end]]) ir = YaoIR(@__MODULE__, src, :qasm_circ) ir.pure_quantum = is_pure_quantum(ir) circ = ZXDiagram(ir) pt_circ = phase_teleportation(circ) 

Here, we got a load a circuit as a ZXDiagram from a .qasm file which can be found here. And we used the phase teleportation algorithm to simplify it. We can see that the T-count of the circuit decreased from 448 to 264.

 123456 julia> tcount(circ) 448 julia> tcount(pt_circ) 264 

The above examples showed how ZXCalculus.jl works as a circuit simplification engine in YaoLang.jl. Now, let’s see what’s behind the scene.

## Low level interfaces of ZXCalculus

In ZX-calculus, we will deal with ZX-diagrams, multigraphs with some extra information. Each vertex of a ZX-diagram is called a spider. There are two types of spiders, the Z-spider and the Z-spider. Each spider is associated with a number called phase. By Dirac notation, the Z-spider and X-spider represent the following rank-2 matrices.

ZX-diagrams can be regarded as a special type of tensor network. On the other hand, quantum circuits can also be regarded as tensor networks. And quantum circuits can be converted to ZX-diagrams according to the following rules. The yellow box, H-box, is just a simple notation of the following spiders in ZX-calculus. To represent general ZX-diagrams, we defined a struct ZXDiagram in ZXCalculus.jl. We can construct a ZX-diagram which represents an empty quantum circuit with n qubits by ZXDiagram(n). For example, if we want to simplify the above circuit with ZXCalculus.jl manually. We first construct a ZX-diagram of a 4-qubits circuit.

 12 using ZXCalculus zxd = ZXDiagram(4) 

Then we can add some gates to the ZX-diagram we have just constructed. We can simply use push_gate! and push_ctrl_gate! to do that.

 123456789101112131415161718192021222324 push_gate!(zxd, Val(:Z), 1, 7//4) push_gate!(zxd, Val(:H), 1) push_gate!(zxd, Val(:X), 1, 1//4) push_gate!(zxd, Val(:H), 4) push_ctrl_gate!(zxd, Val(:CZ), 4, 1) push_ctrl_gate!(zxd, Val(:CNOT), 1, 4) push_gate!(zxd, Val(:H), 1) push_gate!(zxd, Val(:H), 4) push_gate!(zxd, Val(:Z), 1, 1//4) push_gate!(zxd, Val(:Z), 4, 3//2) push_gate!(zxd, Val(:X), 4, 1//1) push_gate!(zxd, Val(:H), 1) push_gate!(zxd, Val(:Z), 4, 1//2) push_gate!(zxd, Val(:X), 4, 1//1) push_gate!(zxd, Val(:Z), 2, 1//2) push_ctrl_gate!(zxd, Val(:CNOT), 3, 2) push_gate!(zxd, Val(:H), 2) push_ctrl_gate!(zxd, Val(:CNOT), 3, 2) push_gate!(zxd, Val(:Z), 2, 1//4) push_gate!(zxd, Val(:Z), 3, 1//2) push_gate!(zxd, Val(:H), 2) push_gate!(zxd, Val(:H), 3) push_gate!(zxd, Val(:Z), 3, 1//2) push_ctrl_gate!(zxd, Val(:CNOT), 3, 2) 

Now, let’s draw the ZX-diagram we have built up. The visualization tool of ZXCalculus.jl is currently provided in YaoPlots.jl.

 12 using YaoPlots plot(zxd) 

We can use the algorithms clifford_simplification 1 and phase_teleportation 2 to simplify this circuit.

 1234 ex_zxd = clifford_simplification(zxd); pt_zxd = phase_teleportation(zxd); plot(ex_zxd) plot(pt_zxd) 

The phase teleportation algorithm can reduce the number of T-gates of a quantum circuit. We can use tcount to show the number of T-gates. In this example, the T-count decreased from 4 to 2.

 123456 julia> tcount(zxd) 4 julia> tcount(pt_zxd) 2 

These algorithms are using the ZX-calculus rules to simplify ZX-diagrams. These rules define how ZX-diagrams are allowed to be transformed. Here are some basic rules for ZXDiagrams. In the paper 1, they defined a special type of ZX-diagram, the graph-like ZX-diagram. We use ZXGraph to represent it in ZXCalculus.jl. And here are some rules for ZXGraphs. One may want to apply rules on a ZX-diagram manually. We provide different APIs for this.

The function match will match all available vertices on a ZX-diagram with a given rule. And we can use the function rewrite! to rewrite a ZX-diagram on some matched vertices. The replace! function just match and rewrite on all matched vertices once. The simplify! function will match and rewrite a ZX-diagram with a rule until no vertices can be matched.

In the clifford_simplification, we will first convert the given ZX-diagram to a graph-like ZX-diagram.

 12 zxg = ZXGraph(zxd) plot(zxg) 

Then we simplify the graph-like ZX-diagram with rule :lc, :p1, and :pab.

 1234 simplify!(Rule{:lc}(), zxg) simplify!(Rule{:p1}(), zxg) replace!(Rule{:pab}(), zxg) plot(zxg) 

Finally, we extract a new circuit from the simplified graph-like ZX-diagram.

 12 ex_circ = circuit_extraction(zxg) plot(ex_circ) 

## Why ZXCalculus.jl?

The above algorithms are first implemented in a Python package PyZX. PyZX is a full-featured library for manipulating large-scale quantum circuits and ZX-diagrams. It provides many amazing features of visualization and supports different forms of quantum circuits including QASM, Quipper, and Quantomatic.

So why we developed ZXCalculus.jl? This is because ZXCalculus.jl is not only a full-featured library for ZX-calculus but also one of circuit simplification engines for YaoLang.jl. A light-weight native implementation of ZX-calculus is necessary since depending on a Python package will make the compiler slower and more complicated.

### Benchmarks

We benchmarked the phase teleportation algorithm on 40 circuits of various numbers of gates (from 57 to 91642). ZXCalculus.jl has 6x to 50x speed-up in these examples (the run time of ZXCalculus.jl is scaled to 1 for each circuit in this picture). These benchmarks are run on a laptop with Intel i7-10710U CPU and 16 GB RAM. The code for benchmarks could be found here. In most examples, the T-count of optimized circuits produced by ZXCalculus.jl is the same as PyZX. However in 2 examples, ZXCalculus.jl has more T-count than PyZX. This may be caused by the different simplification strategies between ZXCalculus.jl and PyZX. We will keep investigating it in the future as mentioned in the next section.

Also, YaoLang.jl support hybrid quantum-classical programs. It is possible to optimize hybrid quantum-classical programs with ZXCalculus.jl.

## Summary and future works

During GSoC 2020, I mainly accomplished the following works.

• Representing and manipulating ZX-diagrams with high-performance.
• Implementing two simplification algorithms based on ZX-calculus.
• Adding visualization of ZX-diagrams to YaoPlots.jl.
• Integrating ZXCalculus.jl with YaoLang.jl.
• Adding support of OpenQASM to YaoLang.jl.

There is still something to be polished.

• Finding a better simplification strategy to get lower T-counts.
• Fully support of visualization of the ZXGraph (the plotting script may fail on some ZXGraph with phase gadgets).
• Converting ZX-diagrams to tensor networks without YaoLang.jl.
• The conversion between the YaoIR and the ZXDiagram may cause the circuit different with a global phase. We should record this global phase in the later version.

Also, I will keep working on YaoLang.jl with Roger Luo to support more circuit simplification methods (template matching methods, Quon based methods, etc.).

## Acknowledgement

I want to appreciate my mentors, Roger Luo, and Jinguo Liu. Without their help, I couldn’t accomplish this project. ZXCalculus.jl is highly inspired by PyZX. Thank Aleks Kissinger and John van de Wetering, the authors of PyZX. They gave me useful advice on the phase teleportation algorithm and reviewed the benchmarks between PyZX and ZXCalculus.jl. Thank Google for holding the Google Summer of Code, which promotes the development of the open-source community.