Noncomputational States¶

Pauli noise acts within a qubit's two-dimensional computational subspace. Real hardware can instead drive the state out of that subspace through leakage, or lose the physical carrier from its site entirely through loss. Neither process is a Pauli channel, so the site no longer holds an ordinary qubit state.

Clifft models these processes with a hybrid quantum-classical trajectory model. Within the computational subspace, the state keeps its full coherent dynamics and entanglement. A leaked or lost site instead has a definite, classically tracked occupation on each trajectory. Transitions between the two are stochastic quantum jumps, including their back-action on the computational state.

This page explains the model and how it composes with Clifft's factored-state simulation. The Leakage and Loss guide shows the Python API.

The effective five-level site¶

Each circuit wire denotes a fixed physical site. For modeling leakage and loss, Clifft uses the effective per-site space

with

The table below lists these levels and their categories:

Under the model's incoherent-jump assumption, there is no coherence between \(\mathcal H_C\) and \(\mathcal H_N\), or between different noncomputational levels. The labels g and e name the computational basis levels used as matrix indices. They do not imply that a computational site occupies a definite level: its state may be any superposition or entangled state in \(\mathcal H_C\).

Each trajectory therefore combines Clifft's ordinary factored quantum state over the computational sites with a classical status ledger over all sites. The ledger records one computational status rather than separate g and e occupations. Its other entries are leak_g, leak_e, and lost, each of which names one definite noncomputational level. The model's initial distribution is sampled independently for each site. An outcome of g or e prepares the corresponding computational basis state; any other outcome sets that definite noncomputational status.

Transitions and measurement classification¶

A transition matrix \(T[\mathrm{to}][\mathrm{from}]\) attaches stochastic jumps to circuit positions. An entry gives the probability of a jump from one source level to one destination level. Each source column must sum to at most one. For any source \(s\), the total jump and no-jump probabilities are

Every nonzero matrix entry is a distinct jump. A diagonal entry \(T[s][s]\) still projects onto its source level; it is not part of the no-jump branch. For a computational source, the column defines a quantum instrument whose outcomes update the live state. For a noncomputational source, it defines an ordinary classical transition from an already definite level.

A measurement classifier \(P[\mathrm{symbol}][\mathrm{level}]\) defines the recorded result for each level. It has two record symbols, 0 and 1, and may have a third herald symbol. If the site occupies leak_g when measured, for example, the leak_g column gives the probabilities of reporting 0, reporting 1, or emitting a herald.

For M and MR on a computational site, the quantum measurement first resolves g or e; the corresponding classifier column can then model computational-basis readout confusion. Computational MX, MY, MRX, and MRY measurements use their ordinary quantum results without applying the g or e columns. Once a site is noncomputational, the classifier supplies the result regardless of measurement basis. The optional herald probability may be nonzero only for noncomputational levels.

Jump back-action¶

A jump from a computational source changes the quantum state, even if its destination is also computational. For a source \(s \in \{g,e\}\) and destination \(\ell\), the transition-matrix entry corresponds to the jump operator

Applying this operator resolves the source against the live coherent state and projects it onto g or e. A computational destination prepares the corresponding basis state and leaves the site in the coherent simulation. A noncomputational destination removes the site from that simulation and records its definite level in the status ledger. This hidden collapse writes no visible measurement record.

For an entangled site, the same collapse updates its partners. Each trajectory keeps the partner state conditioned on the selected jump outcome; averaging over trajectories recovers the correct reduced-state statistics. A Bell-pair partner, for example, is maximally mixed in the ensemble after source-independent loss of the other half.

When no jump occurs, the state is also updated. If the total jump rates from g and e are \(p_g\) and \(p_e\), the no-jump outcome applies

The state is renormalized after conditioning on this outcome. Equal rates make \(K_{\mathrm{stay}}\) proportional to the identity; unequal rates change the surviving coherent state.

A jump from a noncomputational source is a classical status change. A destination of g or e returns the site to the coherent simulation, prepared at that computational basis level.

After a site becomes noncomputational¶

Once a site has a definite noncomputational level, it no longer participates in coherent evolution. Under the current policy, most operations that touch it cannot act, a single-site measurement samples its result from the classifier without regard to measurement basis, and a reset or later transition may restore it to the computational subspace. The Leakage and Loss guide defines the exact behavior for supported circuit operations.

The visible binary result occupies the same record slot as the original measurement. Later rec references, detectors, observables, and classical feedback all consume that substituted bit. When the classifier emits its third symbol, a separate herald marks the slot and the binary record receives a uniformly drawn placeholder. The herald identifies the readout, not the time or location of the underlying jump, so it is not an exact spacetime erasure flag.

How this composes with Clifft¶

Ordinary Clifft compiles a circuit once and reuses the program for many shots. The compiler absorbs deterministic Clifford evolution into an offline frame, plans active stabilizer coordinates, and prepares fixed actions for the symbolic-coordinate executor.

With noncomputational transitions, the sampled history can change which later operations act, which measurements use the classifier, and when a site returns to the computational subspace. Those choices differ between shots, so one program compiled before sampling cannot describe every trajectory.

noncomp.sample instead alternates execution with compilation when the sampled history requires it. The executor directly handles outcomes whose state update leaves the remaining program valid. When an outcome invalidates the compiled remainder -- for example, when a computational site becomes noncomputational -- the executor stops at that transition and returns control to the trajectory driver. The driver samples any destination the executor did not already select, records the outcome, updates the status ledger, rewrites the original circuit, and compiles a continuation. The continuation preserves the prefix already executed, changes the remaining operations to match the new trajectory, and resumes after the transition. Jumps whose source is already noncomputational are sampled as classical status changes while this continuation is constructed.

The prepared transition action holds only the source-dependent total jump rates, the separate weights for g and e destinations, and one combined weight for all noncomputational destinations. This is enough for the executor to evaluate the live computational state using active coordinates. The trajectory driver retains the original five-level matrix so that it can resolve the combined noncomputational outcome outside the executor.

Each continuation still uses Clifft's normal compiler and symbolic-coordinate architecture. Clifford operations, coordinate planning, and symbolic dependencies are resolved before execution resumes.

Active-width cost¶

With the default exact damping policy, the simulation is exact for this hybrid quantum-classical model. Most transition positions do not increase the active width. A source already in the active state uses its existing array axis. A definite dormant source is determined entirely from the Clifford and Pauli frames.

The exceptional case is a coherent dormant site with source-dependent total jump rates, \(p_g \neq p_e\). The \(K_{\mathrm{stay}}\) operator above is then not proportional to the identity, and is non-Clifford. Exact simulation must promote the site from dormant to active.

If this occurs frequently, it can increase the peak active width \(k\), the dominant exponent in Clifft's runtime. Users can instead choose damping="neglect", which omits the no-jump back-action. This is exact when \(p_g = p_e\), when the operator is proportional to the identity. Otherwise it changes the conditioned no-jump state by order \(\lvert p_g-p_e\rvert\) at each transition position. Under this policy, these transition positions do not increase \(k\).