*>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)
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 locally, or explore it here:

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 available by toggling §.

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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! See the GKP example above. CV computing is under active development and not yet fully-fledged, however.

Q. Aren't operators just as low-level as circuits?
A. In the same way that gauge theory is equivalent to regular electrodynamics. The former gives a more abstract way of thinking about the latter.

Q. Are there any concrete complexity-theoretic benefits?
A. Using operators can lead to exponential improvements to simulation; stabilizer circuits/tableaus are a special case.

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 (braided presentations).

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. Since global phase is fake anyway and a source of Heisenbugs, it's no great 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 governed 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, along with further exercises.

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D. Wakeham, Structure and Interpretation of Quantum Programs II: The Way of Yaw. Torsor Labs, 2026. yawlang.dev, accessed on {insert.date}.

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