*>yaw

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Grover search, with database size \(d\) a functional parameter:

def grover(d, marked):
    $alg = qudit(d)
    def reflect(P): return I - 2*P
    G = reflect(proj(Z, marked)) * (-reflect(proj(X, 0)))
    r = int((pi/4)*sqrt(d)) + 1
    return measure(Z, (G**r) « char(X, 0))()[1]

Checking Knill-Laflamme for the five-qubit code:

alg   = qudit(2, n=5); X, Z, I = alg.X, alg.Z, alg.I
stabs = [cycle(k, X[0, 3]*Z[1, 2]) for k in range(4)]
P     = prod(((@5)I + S)*0.5 for S in stabs)
errs  = [E[k] for k in range(5) for E in (X, Z)]
def kl(A, B): return scalar(P * A.d * B * P, P)
all(kl(A, B) is not None for A in errs for B in errs) # True

Defining the GKP state on a photonic quantum computer:

$alg = <X, Z | unit, braid(-1)>
gkp = weylState(lambda a, b: 1 if a % 2 == 0 else 0)
gkp | X**2, gkp | Z**2 # (1, 1)
gkp | X, gkp | Z       # (0, 1)

See SIQP II for more examples.

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Install the repo (requires Python 3.10 or later):

git clone https://github.com/torsor-io/yaw.git
cd yaw
pip install -r requirements.txt

Run the REPL:

python yaw/yaw_repl.py

Compile a .yaw file:

python yaw/y2py.py program.yaw -o program.py
python program.py

Import as package:

from yaw import *

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This site hosts Structure and Interpretation of Quantum Programs II: The Way of Yaw, a living textbook which develops the language from scratch and uses it to build a canonical set of quantum algorithms. A live table of contents is on the left.

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Q. Isn't this just a library with fancy syntax?
A. No, it's a different way of thinking about quantum computing and a language to implement that thinking.

Q. Can I keep working with qubits?
A. Sure, but you address them via their Pauli algebra and corresponding eigenvectors rather than bitstrings.

Q. Can I do continuous variable computing?
A. Yes! This is a neat generalization of the algebra of qudits.

Q. Aren't operators just as low-level as circuits?
A. Yes, in the same way gauge theory is equivalent to regular electrodynamics. The former abstracts the latter.

Q. Presentations are related to the word problem, which is undecidable, right?
A. Yes, so we restrict to a decidable fragment that happens to include everything we care about.

Q. How can you actually compile any of this to hardware?
A. Cramming one Hilbert space into another is called error correction, and the tools for it already exist. Implementation is a work in progress.

Q. Doesn't your approach throw away phase information?
A. Global phase yes, relative phase no. Global phase is fake anyway (and a source of bugs) so it's no loss.

Q. Sounds like a lot of hard math…
A. You can get by with high school algebra! Operators are variables like \(x\) and \(y\) and their behaviour is given by equations.

Q. What does "yaw" mean?
A. It is a self-conjugating recursive acronym for "Yaw Algebraic Way". It's also the direction you head.

Q. Where are the solutions to exercises?
A. Coming to the appendix soon!

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D. Wakeham, Structure and Interpretation of Quantum Programs II: The Way of Yaw. Torsor Labs, 2026. https://yawlang.dev

D. Wakeham, Structure and Interpretation of Quantum Programs I: Foundations. Torsor Labs, 2025. arXiv:2509.04527.