feat: Quil arithmetic expressions as gate arguments in the exact simulators

[bramathon]

Mini-PR onto 1855-experimental-exact-simulators (#1860), as requested in the review discussion there.

What

A gate angle may now be any Quil arithmetic expression over memory references: + - * / ^, the functions SIN, COS, SQRT, EXP, CIS, and real or complex literals. RX(theta[0]/2 + pi) (what quilc emits for a compiled parametric program), RZ(2*SIN(phi[1])) and CPH(CIS(theta[0])) 0 for a DEFGATE with a complex parameter all simulate directly instead of failing at construction.

How

• _resolver._compile_expression walks the pyQuil expression tree and produces a ParameterExpression: the slots it reads, a JAX closure, and a shape key with references replaced by %0, %1, … (SIN(theta[0]) and SIN(theta[1]) share the key SIN(%0)). ParametricGate.arguments holds a literal or a ParameterExpression per gate argument.
• _GateBatch keys on (gate, literals, expression shapes, embedding) and gathers each member's slot values, so same-shaped expressions still share one vmap; the traced graph stays proportional to the number of distinct gate kinds.
• Complex-valued arguments (CIS, a complex literal) are accepted for DEFGATE gates and rejected with a clear error for the built-in gates, which take real angles (quax's gate constructors reject complex angles). SQRT/^ are evaluated in real arithmetic; a negative radicand gives nan where Quil would give a complex number, which the docstring says.
• Parameter-free expressions such as SIN(pi/4) fold to literals.

Tests

test/unit/test_parameter_expressions.py (48 tests): shape keys and slot order for 13 forms, key sharing across slots, the complex flag, numeric evaluation against Quil semantics (including CIS and complex literals), equality with the literal gate for 12 expression shapes, a quilc-style Quil-text program, CIS/complex arithmetic through a DEFGATE, the complex-argument error for built-in gates, feed-forward still rejected, slot sharing/batching, jit + grad through RX(SIN(theta)) (analytic derivative), and literal folding.

🤖 Generated with Claude Code

https://claude.ai/code/session_01JfZKmgqinMhR4BS87y4G4F

[github-actions]

Bencher Report

Project	pyquil
Branch	1860-affine-parameters
Testbed	ci-runner-linux

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Benchmark	Latency	seconds (s)
test/benchmarks/test_program.py::test_copy_everything_except_instructions	📈 view plot ⚠️ NO THRESHOLD	9.61 s
test/benchmarks/test_program.py::test_instructions	📈 view plot ⚠️ NO THRESHOLD	3.35 s
test/benchmarks/test_program.py::test_iteration	📈 view plot ⚠️ NO THRESHOLD	3.31 s

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