Theoretical Overview¶

Clifft is a compiler and execution engine for exact simulation of universal quantum circuits. For circuits where non-Clifford effects remain localized, it confines exponential work to an active-state dimension of \(2^k\), set by the dynamic active width \(k\), instead of the total physical-qubit count.

Method Provenance¶

The original Clifft design established the factored active-state model, described by Bradley A. Chase and Farrokh Labib in the Clifft paper.

SymFT, by Wang Fang, Huazhe Lou, and Riling Li, describes itself as the second-generation successor to SOFT. Its planner builds on SOFT's generalized-stabilizer simulation and Clifft's dense active-state representation. SymFT adds symbolic Clifford-Pauli-frame factorization, adaptive stabilizer-coordinate planning, and direct multi-coordinate kernels.

The current Clifft sampler adopts these SymFT developments. SamplingPlan, host-specific executable preparation, instruments and continuations, and the executor organization remain Clifft-specific implementation choices.

See Symbolic Sampling in Clifft for the release-oriented migration history and matched performance comparison.

Symbolic Clifford Coordinates¶

Sampling a circuit with noise or mid-circuit measurements produces a trajectory. After planned step \(j\), let \(s_{\le j}\) denote the Boolean outcomes sampled so far. The resulting pure state has the factorization

\[ |\psi_j(s_{\le j})\rangle \sim C_j\, P_j(s_{\le j})\, \Big( |\phi_j(s_{\le j})\rangle_{A_j} \otimes |0\rangle_{D_j} \Big). \]

Here \(\sim\) denotes equality up to global phase. The represented state and the active coefficient array are normalized.

The factors have distinct roles:

\(C_j\) (Clifford coordinate map): Maps the current stabilizer coordinates into the physical qubit basis. The front end and planner determine its evolution before sampling; a shot performs no tableau operations.
\(P_j(s_{\le j})\) (Pauli frame): Represents branch-dependent Pauli corrections from noise, measurement outcomes, and classical feedback. It is a mathematical component of the trajectory state, but the sampling executor does not materialize it as a mutable \(n\)-qubit runtime frame.
\(A_j\) and \(D_j\) (active and dormant coordinates): Partition the current stabilizer coordinates into an ordered active prefix of width \(k_j\) and \(n-k_j\) dormant coordinates. Dormant coordinates are stabilized in the computational zero state in this basis.
\(|\phi_j\rangle_{A_j}\) (active coefficient state): A dense complex array of size \(2^{k_j}\) containing the non-Clifford interference. Ordinary sampling keeps this coefficient state normalized up to floating-point drift.

Active and dormant coordinates are basis elements, not subsets of physical qubits. After Clifford gates change the basis, one coordinate may represent a different, possibly multi-qubit, physical Pauli without changing the size of the active array. Throughout these docs, active width \(k_j\) means the number of stabilizer coordinates represented in the dense coefficient array. The corresponding active-state dimension is \(2^{k_j}\), the number of amplitudes in that array. Neither is the number of physical qubits touched by the circuit.

For a normalized physical circuit, the unconditional noisy state is the ensemble over trajectories,

\[ \rho_j = \mathbb{E}_{s_{\le j}} \left[ |\psi_j(s_{\le j})\rangle \langle\psi_j(s_{\le j})| \right]. \]

Clifft samples members of this ensemble rather than materializing the full density matrix.

Why the Factorization Matters¶

The dense active array is the only state component with exponential size. If \(k_{\max}\) is the largest active width reached by a plan, coefficient and scratch storage scale as \(O(2^{k_{\max}})\) instead of \(O(2^n)\). Frequent measurements can return coordinates to the dormant set, so fault-tolerant circuits may retain a small \(k_{\max}\) even when they contain hundreds of physical qubits.

Planning Pauli-Frame Effects as Symbolic Signs¶

The main symbolic idea is to resolve the branch-dependent Pauli frame during planning instead of updating it after every event in every shot. Following the symbolic-frame strategy introduced by Fang, Lou, and Li, the planner expresses each relevant dependence as an affine formula of Boolean symbols and attaches the resulting sign to the affected operation.

Each stochastic event needed later in a shot receives a Boolean symbol. A symbol may represent a presampled Pauli fault, a sampled measurement branch, a readout flip, or an instrument outcome. These effects are represented as affine Boolean expressions,

\[ \ell(s) = c \oplus \bigoplus_{r \in R} s_r. \]

For any Pauli observable or operation \(Q\), conjugation by the trajectory's Pauli frame can change only its sign:

\[ P_j(s)^\dagger Q P_j(s) = (-1)^{\ell_Q(s)} Q. \]

The planner computes \(\ell_Q\) once and maps the unsigned body of \(Q\) into the current stabilizer coordinates. An operation that needs coefficient work is therefore represented by an active-coordinate Pauli and an affine sign, rather than by a physical Pauli string plus a mutable runtime frame.

As a result, a shot does not carry and update an \(n\)-qubit Pauli frame after each fault or measurement. It evaluates the prepared expressions as their symbols become available and uses the realized signs when applying rotations, measurements, and output actions.

Adaptive Stabilizer Coordinates¶

The planner maintains a stabilizer-destabilizer basis in which the first \(k_j\) coordinates are active and the remainder are dormant. In this basis, each dormant coordinate is in \(|0\rangle\) and has \(Z\) as its stabilizer. A promotion moves a dormant coordinate into the dense state when it must carry coherent amplitudes; a planned measurement and collapse can return an active coordinate to the dormant set.

The planner resolves these changes once for all shots. Some representative cases are:

Situation	Example	Planned state action	Width
Rotation already supported on active coordinates	A mapped \(Z\) rotation that touches only active coordinates	Rotate the active Pauli directly	\(k \to k\)
Rotation requiring dormant coherent support	An \(X\)-axis pi/4 rotation on a dormant \(\lvert 0\rangle\) coordinate	Promote the coordinate, then rotate	\(k \to k+1\)
Measurement with active support	A mapped \(Z\) measurement that touches the active state	Sample and collapse, then remove the chosen coordinate	\(k \to k-1\)
Random measurement in dormant space	An \(X\) measurement of a dormant \(\lvert 0\rangle\) coordinate	Replace its stabilizer and define the sampled branch	\(k \to k\)
Classical result	A deterministic \(Z\) record or a detector parity of earlier records	Update only symbols, records, or outputs	\(k \to k\)

This is Clifft's adaptation of SymFT's adaptive stabilizer-coordinate planning. For an active measurement, the planner chooses a basis and array dimension that can be removed after collapse. A random dormant measurement changes the stabilizer description and symbolic correction without traversing the dense coefficient array.

Transition instruments use the same state model. They may act classically, filter or collapse the active state, promote a coordinate, or stop execution so that a trajectory-specific continuation can be prepared.

From a Circuit to Samples¶

The boxes below name the main compiler and runtime objects. The text between them describes the work that prepares or consumes each object.

[Circuit text]
      |
      | Parse and absorb Clifford gates
      v
[Heisenberg IR]
      |
      | Fuse, cancel, and simplify Pauli operations
      v
[Optimized HIR]
      |
      | Choose coordinates and derive symbolic dependencies
      v
[SamplingPlan]
      |
      | Prepare fixed storage, fusion, and scalar or SIMD kernels
      v
[ExecutablePlan]
      |
      | Allocate reusable shot state
      v
[Executor]
      |
      | Run shots and collect outputs
      v
[Records, detectors, observables, and other results]

Clifford Trace¶

The front end turns circuit text into the operations Clifft needs to simulate. It absorbs physical Clifford gates into an offline tableau and maps rotations, measurements, noise, feedback, detectors, and observables into the Heisenberg basis. These operations form the Heisenberg IR, or HIR.

HIR Optimization¶

HIR passes use Pauli algebra and dataflow to fuse or cancel operations and, when safe, shorten how long coordinates must remain active. This reduces the work handed to the planner without fixing a runtime representation.

Coordinate Planning¶

The symbolic-coordinate planner decides which coordinates are active at each step. It also resolves basis changes, Pauli support, measurement collapse, and the symbolic signs described above. The result is a SamplingPlan that gives every shot the same semantic sequence of possible actions.

SamplingPlan is the executor-independent semantic boundary. It describes symbols, affine expressions, active-coordinate actions, width transitions, records, outputs, noise sites, instruments, and continuation boundaries. It does not select an ISA, SIMD kernel, descriptor layout, or target-specific fusion.

Executable-Plan Preparation¶

Preparation turns the semantic plan into fixed storage for the selected executor backend. It arranges symbolic dependencies for incremental evaluation, combines supported rotation runs, and selects scalar or architecture-specific kernels. On x86 builds with runtime dispatch, this selects the scalar, AVX2, or AVX-512 backend once for the plan. Apple arm64 builds use NEON kernels; other CPU and WebAssembly builds use the scalar backend. The experimental HIP backend instead prepares a backend-private GPU executable from the same semantic plan.

Sampling¶

The executor allocates coefficient, scratch, symbol, record, output, and RNG storage before the hot loop, then reuses the prepared plan and that storage for every shot. It samples fault and measurement symbols, evaluates the affected expressions, and applies prepared active-state actions. It performs no tableau evolution, commutation analysis, Pauli localization, coordinate selection, or dependency discovery.

Continuations and Noncomputational Trajectories¶

A noncomputational transition, such as leakage or loss, can change whether later gates still act quantum mechanically on a site. Clifft therefore places explicit boundaries after relevant instruments. A boundary is a prepared point where ordinary execution may stop so that the remaining circuit can be rewritten for that trajectory's sampled site status.

Clifft then plans a continuation and resumes. The continuation preserves the live coefficients, coordinate meaning and order, active width, symbol and record values, and RNG position. The same factored trajectory state therefore spans the boundary even though the replacement suffix is planned later.

See Noncomputational States for the hybrid quantum-classical model.

Final-State Queries¶

For eligible pure-state plans, Clifft retains the final Clifford coordinate map needed to relate the active coefficient state back to physical qubits. This metadata is not read by ordinary sampling dispatch. It supports exact queries such as dense statevector expansion and sparse computational-basis probabilities.

Dense statevectors preserve relative amplitudes and phases but are defined only up to global phase. Basis and record probabilities are exact and are invariant under that choice of statevector representative.

clifft.basis_probabilities() computes selected full-register probabilities without expanding the full \(2^n\) statevector. Its exponential component scales as \(2^k\): exponentially in the active width \(k\), or linearly in the active-state dimension.

For pure-state programs whose only stochastic events and outputs are visible measurements, and which do not use postselection, clifft.record_probabilities() evaluates selected records exactly without sampling.

See Basis-State Probabilities and the Exact Probabilities guide.

References¶

Bradley A. Chase and Farrokh Labib, "Clifft: Fast Exact Simulation of Near-Clifford Quantum Circuits," arXiv:2604.27058, 2026.
Wang Fang, Huazhe Lou, and Riling Li, "SymFT: Universal Fault-Tolerant Quantum Circuit Simulation via Symbolic Clifford-Pauli Frames and Stabilizer Coordinates," arXiv:2607.28600 (quant-ph), 2026.