Bosonic and Encoded Computation Models

A bosonic encoded computation uses the infinite-dimensional state space of one or more oscillator modes to carry a declared finite logical system. Its logical operation is not obtained by naming a physical pulse or projecting each pulse separately: it is induced by the complete ordered physical program, including measurements, recovery, accepted records, and any tracked frame. The central task is therefore to expose the map from physical dynamics to logical output while retaining leakage, rejection, normalization, and resource costs.

This page develops that audit for ideal and noisy programs. It does not derive bosonic code families, fault-tolerance thresholds, direct continuous-variable algorithms, device controls, or experimental performance. Those are separate questions even when they use the same oscillator.

Required background. Bosonic Codes supplies the channel–code–recovery framework, Knill–Laflamme conditions, syndrome and decoder concepts, and the main code families. Circuit Model supplies register, channel, instrument, classical-record, adaptive-composition, and input–output grammar.

Oscillator-Encoded Computation as an Induced Logical Model

For $m$ oscillator modes, fix the algebra

$$[a_j,a_k^\dagger] = \delta_{jk}, \qquad [a_j,a_k] = 0,$$

and the total number convention

$$N_{\mathrm{tot}} := \sum_{j=1}^{m} a_j^\dagger a_j.$$

These relations do not yet define a computation. A finite logical register also requires an encoding, a physical alphabet, a preparation rule, an ordered program, an output decoder, and a success criterion. Unbounded generators require a stated common invariant domain, such as the finite-particle span, or a declared finite-energy input family on which every displayed product is defined.

The relevant hierarchy is

$$\text{logical input} \xrightarrow{\operatorname{Enc}} \text{physical program} \xrightarrow{\text{instrument and recovery}} \text{decode and flag} \xrightarrow{\text{metric}} \text{licensed claim}.$$

Only this complete composition defines the induced logical model. A physical Gaussian unitary need not become a Gaussian logical gate after restriction to a finite code, and a non-Gaussian code state does not by itself confer universality. Likewise, code preservation is weaker than error correction, and exact ideal algebra says nothing by itself about fault propagation, hardware fidelity, or computational advantage.

This distinction has been central since early loss-adapted bosonic encodings and oscillator codes Chuang, Leung, and Yamamoto (1997) and Gottesman, Kitaev, and Preskill (2001). The present purpose is narrower: given an encoding and physical program, determine exactly what logical operation that program licenses.

The Ten-Field Bosonic Encoded Computation Record

Use the following vertical record for every worked program. A field that does not apply must say N/A and explain why; silence is not a resource convention.

1. Computational task, input family, and licensed claim — Declare the logical input family, requested transformation or output, and whether the claim is fixed-input, channel-wide, exact, approximate, conditional, or fault tolerant.
2. Mode register, encoding isometry, code projector, basis, and energy convention — Name the modes, commutators, number convention, isometry, projector, logical basis, finite-energy family, and any superselection promise.
3. Physical controls, noise, measurements, domains, and promises — Give every unitary, channel, instrument, calibrated parameter, idealization, and domain used by the physical description.
4. Preparation, ancillas, recovery, and logical-frame convention — State how data and ancillas are prepared, which recovery is physically applied, which logical correction is tracked, and which preparation costs are counted.
5. Program order, classical record, adaptivity, and accepted branches — Freeze composition order, measurement outcomes, causal dependencies, retries, the accepted record set, and the stopping rule.
6. Induced logical map, leakage, rejection, and normalization — Compose the physical program first, then report the decoded flagged map, code acceptance, leakage or rejection, and any input-dependent normalization.
7. Output, decoder, target, and error metric — Identify the output system, decoder, target object, accepted event, and comparison metric; keep conditional and unconditioned quantities distinct.
8. Resources, fault assumptions, comparator, and owner boundary — Count modes, energy, cutoffs, controls, depth, ancillas, measurements, bits, feedforward, recovery, acceptance, and repeats, then justify every N/A and name the comparator.
9. Verification data, truncation control, uncertainty, and reproducibility — Supply exact identities, frozen numerical values, arithmetic precision, tolerances, a truncation certificate, an independent evaluation route, and uncertainty or its justified N/A.
10. Conclusion, stopping point, and canonical handoff — State only what the record establishes, what it does not establish, and which specialist owner receives the remaining question.

The record is intentionally longer than a gate name. It prevents an attractive code-space matrix from hiding the physical branch structure and prevents a resource claim from silently treating projection, recovery, reset, or retry as free.

Encoding Isometries, Logical Projectors, and Finite-Energy Inputs

Let the logical Hilbert space have dimension $d_{\mathrm L}$ and let

$$W: \mathbb C^{d_{\mathrm L}} \longrightarrow \mathcal H_{\mathrm{phys}}$$

be an isometry. Define

$$W^\dagger W = I_{\mathrm L}, \qquad P := WW^\dagger, \qquad Q := I-P,$$

and the encoding channel

$$\operatorname{Enc}(\rho) := W\rho W^\dagger.$$

The ordered logical basis is part of the specification: changing it changes the matrix assigned to a physical operation. So is the energy convention. For example, a useful constrained family is

$$\mathfrak F_E := \left\{ \rho: \operatorname{Tr}(\rho N_{\mathrm{tot}}) \leq E \right\}.$$

A hard occupation cutoff and a mean-energy constraint are not interchangeable. A code with finite Fock support can still be acted on by a physical operation with infinite Fock support. Conversely, an unbounded generator may be well-defined on the finite-particle invariant domain even when it has no bounded operator norm on the full Hilbert space.

The projector $P$ identifies code weight, but $P\sigma P$ is generally subnormalized. Projection is a physical measurement only when an instrument that produces its outcome is included. It is not automatically a recovery, nor may discarded weight be restored by renaming the surviving state as the logical output.

Code construction, error sets, syndrome logic, and recovery design remain at Bosonic Codes. The carrier-versus-encoding distinction remains at Bits, Qubits, Qudits, and Modes.

Physical Programs and Induced Logical Channels

For a physical unitary $U$, define its code block and leakage block by

$$M_U := W^\dagger U W, \qquad L_U := Q U W.$$

Unitarity gives the exact conservation identity

$$M_U^\dagger M_U + L_U^\dagger L_U = I_{\mathrm L}.$$

For a normalized logical vector $|\psi\rangle$, the leakage probability and its worst-case version are

$$p_{\mathrm{leak}}(\psi;U) := \lVert L_U|\psi\rangle \rVert^2,$$ $$p_{\mathrm{leak}}^{\max}(U) := \lVert L_U\rVert_{\mathrm{op}}^2.$$

The physical unitary realizes a target logical unitary $V$ exactly, up to a global phase, only when

$$UW = e^{i\alpha}WV$$

on the declared input family. When a projected code block is compared with a target, one useful phase-insensitive quantity is

$$d_{\mathrm{proj}}(M_U,V) := \min_{\alpha\in\mathbb R} \left\lVert M_U-e^{i\alpha}V \right\rVert_{\mathrm{op}}.$$

It must be reported alongside leakage. A small distance after conditioning does not make a leaking operation deterministic.

Most importantly, projection does not commute with composition. For two physical unitaries,

$$\begin{aligned} W^\dagger U_2U_1W =\;& M_{U_2}M_{U_1} \\ &+ W^\dagger U_2 Q U_1W. \end{aligned}$$

The second term records amplitude that leaves the code after $U_1$ and returns under $U_2$. Multiplying the projected blocks step by step discards that path and can give the wrong logical program even when the final physical unitary preserves the code.

For a general physical channel $\Phi_{\mathrm{phys}}$, retain lost code weight with the flagged decoder

$$\begin{aligned} \mathcal D_\perp(\sigma) :=\;& W^\dagger P\sigma P W \\ &+ \operatorname{Tr}(Q\sigma) |\perp\rangle\langle\perp|. \end{aligned}$$

The induced flagged logical channel is then

$$\mathcal E_{\mathrm L} := \mathcal D_\perp \circ \Phi_{\mathrm{phys}} \circ \operatorname{Enc}.$$

Here $\Phi_{\mathrm{phys}}$ means the complete physical program, not one selected pulse. Generic channel and instrument composition belongs to the Circuit Model; this page owns the oscillator-encoding specialization.

Measurements, Recovery, and Adaptive Logical Frames

Suppose an outcome $y$ has physical measurement Kraus operators $K_{y\mu}$, followed by a licensed physical recovery with operators $R_{y\nu}$. Let $F_y$ be the logical-frame unitary that maps the decoded branch to the declared canonical frame. First define the raw decoded branch operators

$$B_{y\nu\mu} := W^\dagger P R_{y\nu}K_{y\mu} W.$$

For an accepted record set $\mathcal A$, three maps encode three different frame conventions. If the accepted record is forgotten and no frame is applied, the raw map is

$$\widetilde{\mathcal E}^{\mathrm{raw}}_{\mathcal A}(\rho) := \sum_{\substack{ y\in\mathcal A\\ \nu,\mu }} B_{y\nu\mu} \rho B_{y\nu\mu}^\dagger.$$

If the frame is tracked rather than applied, retain $y$ in a classical register $C$ together with the declared table $y\mapsto F_y$:

$$\begin{aligned} \widetilde{\mathcal E}^{\mathrm{track}}_{\mathcal A}(\rho) :=\;& \sum_{y\in\mathcal A} |y\rangle\langle y|_C \\ &\otimes \sum_{\nu,\mu} B_{y\nu\mu} \rho B_{y\nu\mu}^\dagger. \end{aligned}$$

No physical $F_y$ is applied in this representation. If every accepted branch is instead brought to the canonical frame before its record is discarded, define

$$A_{y\nu\mu} := F_yB_{y\nu\mu},$$ $$\widetilde{\mathcal E}^{\mathrm{can}}_{\mathcal A}(\rho) := \sum_{\substack{ y\in\mathcal A\\ \nu,\mu }} A_{y\nu\mu} \rho A_{y\nu\mu}^\dagger.$$

Operator order is operational. The physical instrument acts first, the physical recovery acts next, and only then does the final projection and decode occur. In the canonical map, $F_y$ acts after decoding because it is the leftmost factor. Tracking a frame means retaining its classical record for a later command or decoder; it does not mean that a physical correction was applied. Raw, tracked, and canonically corrected outputs must not be silently identified.

The record $y$ can contain an integer syndrome, an analog value, a sequence of ancilla outcomes, or a leakage flag. The accepted record set must be fixed before inspecting the desired answer. If a later control depends on $y$, the physical branch including that control is composed before decoding. A readout label alone is not a recovery map, and a recovery is not licensed merely because a code family admits one in principle.

The formal theory of outcome-resolved maps is developed at Quantum Instruments. Bosonic syndrome interpretation and decoder design remain with Bosonic Codes and Decoders.

Leakage, Rejection, and Conditional Normalization

The raw, tracked, and canonical accepted maps have the same probability:

$$p_{\mathcal A}(\rho) := \operatorname{Tr} \widetilde{\mathcal E}^{\mathrm{raw}}_{\mathcal A}(\rho)$$ $$\begin{aligned} p_{\mathcal A}(\rho) &= \operatorname{Tr} \widetilde{\mathcal E}^{\mathrm{track}}_{\mathcal A}(\rho) \\ &= \operatorname{Tr} \widetilde{\mathcal E}^{\mathrm{can}}_{\mathcal A}(\rho). \end{aligned}$$

Only if $p_{\mathcal A}(\rho)>0$ may the corresponding conditional output be defined. For example, the canonical-frame conditional state is

$$\rho_{\mathcal A}^{\mathrm{can}} := \frac{ \widetilde{\mathcal E}^{\mathrm{can}}_{\mathcal A}(\rho) }{ p_{\mathcal A}(\rho) }.$$

Raw and tracked conditional outputs are defined by dividing their respective subnormalized maps by the same probability. When $p_{\mathcal A}(\rho)=0$, every conditional output is undefined, not the zero state.

These quantities answer different questions:

• Leakage is physical weight outside the declared code projector.
• Rejection is probability assigned to records outside the accepted set.
• Flag probability is weight delivered to an explicit orthogonal output.
• Conditional error compares a normalized accepted output with its target.
• Unconditioned error retains the probability cost of rejected or flagged outcomes.

Leakage and rejection coincide only when the instrument explicitly rejects exactly the $Q$ branch. A protocol may accept leaked states, reject an in-code syndrome, or recover leakage before decoding. The equality must therefore be derived, not assumed.

When $p_{\mathcal A}$ depends on $\rho$, normalization makes each chosen conditional rule nonlinear. Such a rule is not a trace-preserving logical channel. Report the chosen frame convention, its linear subnormalized accepted map, the rejected or flagged branch, and conditional and unconditioned metrics together. Retrying is licensed only when the protocol supplies a fresh known input, a recoverable backup, or another explicit restart mechanism.

Native Logical Gates, Universality, and Completion Resources

A physical control is an exact native logical gate only if it preserves the code and realizes the target on the full declared logical input family. An approximate native gate additionally needs a topology, tolerance, and uniform error statement. A postselected accepted map is a third object: it needs an acceptance function and cannot be called deterministic merely because its normalized output is close to a target.

A native-alphabet record must declare:

• the encoded family and physical controls;
• availability of inverses and parameter precision;
• common domains for unbounded generators;
• leakage, accepted branches, and recovery;
• ancillas, measurements, frames, and reset assumptions;
• the logical target topology and approximation tolerance.

One exact encoded rotation does not prove universality. A family consisting only of diagonal $R_z$ rotations and controlled-$Z$ gates remains diagonal. A completion resource such as a Hadamard, measurement-induced nonlinearity, or code-specific operation must be named and costed. General exact and approximate reachability belongs to Universal Gate Sets, while rewriting targets into a declared alphabet belongs to Gate Decomposition.

Bosonic proposals realize gates by substantially different mechanisms: linear optics and measurement Knill, Laflamme, and Milburn (2001), dispersive oscillator control Krastanov et al. (2015), rotation-symmetric constructions Grimsmo, Combes, and Baragiola (2020), and stabilized cat-code operations Mirrahimi et al. (2014). These results motivate possible alphabets; none permits an unspecified primitive to be imported for free.

Truncation, Energy, and Program Resource Accounting

For a total-number cutoff retaining sectors below $N$, define

$$\Pi_{<N} := \sum_{\sum_j n_j<N} |n_1,\ldots,n_m\rangle \langle n_1,\ldots,n_m|.$$

If

$$\bar n_{\mathrm{tot}} := \operatorname{Tr} \left( \rho N_{\mathrm{tot}} \right),$$

then Markov’s inequality gives

$$\operatorname{Tr} \left[ (I-\Pi_{<N})\rho \right] \leq \frac{ \bar n_{\mathrm{tot}} }{ N }.$$

This certifies an input-state tail only. It is not a unitary, channel, or logical-gate error bound. A truncated simulation must separately state the cutoff convention, boundary treatment, convergence observable or norm, tolerance, and comparison across cutoffs. Finite-matrix convergence does not establish a full-space operator-norm approximation for an unbounded bosonic generator.

Resource accounting should distinguish fixed footprint, per-attempt consumption, and expected cost per accepted result. A compact vertical ledger should include:

• storage modes and simultaneously live ancillas;
• mean occupation and a tail or hard-cutoff convention;
• native controls, completion operations, and physical depth;
• measurements, classical bits, feedforward, recovery, and frame tracking;
• acceptance probability, stopping rule, and expected repeats;
• modeled noise and fault assumptions;
• the reference task and comparator boundary.

An expected number of attempts multiplies consumed controls and measurements, but it does not multiply a reused hardware footprint. Conversely, a small mode count does not make preparation, readout, nonlinear ancillas, controller latency, or energy free. Resource Estimation for Fault-Tolerant Quantum Computing owns system-level spacetime accounting; this page stops at the declared encoded-program ledger.

Correctness Metrics and Evidence Boundaries

Choose the metric to match the licensed output:

• use exact equality up to global phase or $d_{\mathrm{proj}}$ together with leakage for an ideal projected unitary;
• use a reference-sensitive channel metric for an arbitrary logical input, including the orthogonal flag;
• use fidelity or trace distance for a declared fixed state;
• use total variation distance for a classical output law;
• report acceptance and both conditional and unconditioned metrics for a selected branch.

A symbolic identity should be checked independently when numerical values are published. State the arithmetic precision, rounding rule, tolerance, and whether exact algebra or numerics is authoritative. A finite cutoff needs a tail or convergence certificate. Experimental uncertainty is N/A only for an explicitly ideal model; otherwise calibration date, uncertainty, sampling, and drift belong in the evidence record.

Bosonic-control demonstrations such as Heeres et al. (2017), error-corrected encoded gates Reinhold et al. (2020), and bias-preserving operations Puri et al. (2020) answer dated experimental questions under particular devices and protocols. Their existence does not turn an ideal program audit into a hardware claim. Device modules and evidence belong to Bosonic Qubits and Continuous-Variable Platforms; fault spread, thresholds, and decoder performance belong to their dedicated fault-tolerance owners.

Worked Audit: An Exact Dual-Rail Beam-Splitter Gate

This audit instantiates the full record without a Fock truncation.

1. Computational task, input family, and licensed claim — Implement $V_{\mathrm L}=e^{-i\pi X/4}$ on an arbitrary dual-rail logical qubit. On a separate fresh $|0_{\mathrm L}\rangle$ verification copy, audit the output amplitudes and optional rail-number statistics. The licensed statement is a channel-wide exact ideal encoded-unitary claim in the declared total-one sector. The gate boundary ends immediately after $U_B$; the optional verification readout is not part of the induced gate channel.

2. Mode register, encoding isometry, code projector, basis, and energy convention — Use modes $a,b$ with $[a,a^\dagger]=[b,b^\dagger]=1$ and vanishing cross-mode commutators. Define $W|0\rangle=|10\rangle$, $W|1\rangle=|01\rangle$, and $P=|10\rangle\langle10|+|01\rangle\langle01|$. The ordered code basis is $(|10\rangle,|01\rangle)$ and the code is the $N_{\mathrm{tot}}=1$ sector.

3. Physical controls, noise, measurements, domains, and promises — Use the ideal beam-splitter unitary

   $$U_B(\theta) := \exp \left[ -i\theta \left( a^\dagger b+a b^\dagger \right) \right], \qquad \theta = \frac{\pi}{4}.$$

   No noise is modeled. The generator is evaluated on the invariant finite-particle sector.

4. Preparation, ancillas, recovery, and logical-frame convention — The gate accepts an already prepared arbitrary code input. The verification trial separately prepares a fresh $|10\rangle$. There is no ancilla, recovery, postselection, or logical-frame update.

5. Program order, classical record, adaptivity, and accepted branches — Apply one beam splitter and stop: this is the complete gate program, with no measurement, classical record, adaptivity, rejection, or retry. An optional joint rail-number readout may be performed afterward on the separate verification copy; it samples the output but does not extend the gate channel.

6. Induced logical map, leakage, rejection, and normalization — On the code, $a^\dagger b+a b^\dagger=X$, and therefore

   $$M_{U_B} = e^{-i\pi X/4} = \frac{1}{\sqrt2} \begin{pmatrix} 1&-i\\ -i&1 \end{pmatrix}, \qquad L_{U_B} = 0.$$

   Code acceptance is $1$; leakage and rejection are both $0$, so no conditional normalization is used.

7. Output, decoder, target, and error metric — Before any verification readout, the decoded gate output is $V_{\mathrm L}\rho V_{\mathrm L}^\dagger$ for every declared input. On the separate verification input, the quantum output is

   $$U_B|10\rangle = \frac{ |10\rangle-i|01\rangle }{ \sqrt2 },$$

   with optional rail-number probabilities $1/2$ and $1/2$, and $d_{\mathrm{proj}}(M_{U_B},V_{\mathrm L})=0$. If performed, that readout produces verification data rather than a quantum gate output.

8. Resources, fault assumptions, comparator, and owner boundary — For one ideal gate invocation on an already prepared code input, count two modes, one excitation, one beam-splitter interaction, and physical gate depth one. Gate measurements, classical bits, ancillas, recovery, feedforward, rejected attempts, and repeats are zero. A separate optional verification trial adds one fresh known-input preparation, one joint rail-number readout, and one classical bit after the gate; none belongs to the gate-channel ledger. Physical time, calibration, control uncertainty, device noise, fault-tolerant overhead, and a hardware comparator are N/A because the record licenses only the ideal finite-sector operation.

9. Verification data, truncation control, uncertainty, and reproducibility — The frozen logical matrix is

   $$M_{U_B} \approx \begin{pmatrix} 0.707106781186548& -0.707106781186548i\\ -0.707106781186548i& 0.707106781186548 \end{pmatrix}.$$

   Verify $M^\dagger M=I$, $L=0$, and the two optional verification probabilities from the exact expression. Exact symbolic evaluation is authoritative. Independent at-least-30-digit and binary64 evaluations must agree componentwise and in probability to $10^{-12}$. No Fock cutoff is required because the one-particle sector is invariant. Experimental uncertainty is N/A.

10. Conclusion, stopping point, and canonical handoff — The record establishes one exact ideal dual-rail logical rotation and its complete model-level cost. It establishes neither an error-correcting code nor universality, fault tolerance, device fidelity, or advantage. Those questions belong to Bosonic Codes, Universal Gate Sets, fault-tolerance, hardware, and evidence owners.

Worked Audit: A Projected Single-Rail Displacement

This audit shows why a regular physical Gaussian operation need not induce a deterministic logical gate.

1. Computational task, input family, and licensed claim — For the fixed logical vacuum input, apply the physical displacement $D(1/2)$, project onto the single-rail code, and compare the accepted state with $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$. This is a fixed-state, postselected audit, not a channel-wide gate claim.

2. Mode register, encoding isometry, code projector, basis, and energy convention — Use one mode with $W|0\rangle=|0\rangle$, $W|1\rangle=|1\rangle$, $P=|0\rangle\langle0|+|1\rangle\langle1|$, and $Q=I-P$. The ordered basis is $(|0\rangle,|1\rangle)$ and number eigenstates obey $n=0,1,\ldots$.

3. Physical controls, noise, measurements, domains, and promises — Use

   $$D(\alpha) := \exp \left( \alpha a^\dagger-\alpha^*a \right), \qquad \alpha = \frac12 \in\mathbb R,$$

   together with the exact coherent-state expansion

   $$D(\alpha)|0\rangle = e^{-|\alpha|^2/2} \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}} |n\rangle.$$

   The displacement and ideal code-space projection are the only licensed physical operations. The Weyl displacement is defined on the full mode Hilbert space; no noise is modeled.

4. Preparation, ancillas, recovery, and logical-frame convention — Prepare a fresh vacuum. There is no ancilla, physical recovery, or decoded frame update. Projection is a binary measurement, not a free correction.

5. Program order, classical record, adaptivity, and accepted branches — Apply $D(1/2)$, then measure $\{P,Q\}$. Accept only the $P$ record. No adaptive continuation is licensed, and a repeat requires another freshly prepared vacuum.

6. Induced logical map, leakage, rejection, and normalization — The full projected branch operator on the single-rail code is

   $$K_{\mathrm L} := W^\dagger D(1/2)W = e^{-1/8} \begin{pmatrix} 1&-1/2\\ 1/2&3/4 \end{pmatrix}.$$

   It is not proportional to a unitary. For the declared vacuum input,

   $$PD(1/2)|0\rangle = e^{-1/8} \left( |0\rangle+\frac12|1\rangle \right).$$

   Hence

   $$p_{\mathrm{code}} = \frac54 e^{-1/4} = 0.973500978839256,$$ $$\begin{aligned} p_{\mathrm{leak}} &= p_{\mathrm{reject}} \\ &= 1-p_{\mathrm{code}} \\ &= 0.026499021160744. \end{aligned}$$

   Here rejection equals leakage because the protocol rejects exactly the $Q$ outcome. The normalized accepted state is

   $$|\psi_{\mathrm{code}}\rangle = \frac{ 2|0\rangle+|1\rangle }{ \sqrt5 }.$$

7. Output, decoder, target, and error metric — With target $|+\rangle$, the conditional fidelity is

   $$F_{\mathrm{cond}} = |\langle+|\psi_{\mathrm{code}}\rangle|^2 = \frac9{10}.$$

   The unconditioned embedded target overlap is

   $$F_{\mathrm{uncond}} = |\langle+|D(1/2)|0\rangle|^2 = \frac98e^{-1/4} = 0.876150880955330.$$

   A logical-gate distance is N/A because only one input state was audited.

8. Resources, fault assumptions, comparator, and owner boundary — Count one mode, one displacement, one binary projection, one classical accept/reject bit, and mean photon number $\bar n=|\alpha|^2=1/4$. The coherent state has infinite Fock support. Independent fresh-vacuum repetition has expected

   $$\frac{1}{p_{\mathrm{code}}} = \frac45e^{1/4} = 1.027220333350193$$

   attempts. Retry of an unknown logical input is neither free nor licensed. Physical timing, calibration, implementation error, recovery, and fault-tolerant overhead are N/A because none is modeled. A hardware comparator is N/A for the same reason.

9. Verification data, truncation control, uncertainty, and reproducibility — Exact formulas are authoritative. At-least-30-digit and independent binary64 evaluations must reproduce every displayed 15-decimal value to $10^{-12}$. No numerical cutoff is used; convergence follows from the exact coherent-state series. Experimental uncertainty is N/A.

10. Conclusion, stopping point, and canonical handoff — The physical Gaussian displacement has high code acceptance and high conditional fixed-state fidelity, but it is not thereby a deterministic logical gate. Conditional normalization cannot erase rejection or license recovery, universality, hardware performance, or fault tolerance. Those claims stop at the canonical code, gate-set, hardware, and fault-tolerance owners.

Common Failure Modes and Canonical Handoffs

• Replacing a physical program by projected pulse blocks. The omitted $Q$-space cross terms can carry amplitude out of and back into the code. Compose the complete physical program first.
• Conflating a code block with a channel. $W^\dagger U W$ may be nonunitary. Include leakage and, when appropriate, a flagged decoder.
• Reporting only conditional fidelity. Also report the subnormalized accepted map, its probability, the rejected or flagged output, and an unconditioned metric.
• Treating projection or restart as free. Projection is an instrument. Repeating an unknown input requires a licensed backup, erasure mechanism, or re-preparation operation.
• Omitting conventions. An isometry without its ordered basis, projector, energy convention, domains, and physical program order does not determine a reproducible logical map.
• Overclaiming a cutoff. A state-tail bound is not a gate or channel bound, and a finite matrix cannot prove full-space operator-norm control for an unbounded generator.
• Overclaiming universality. One exact gate or a diagonal native family is insufficient; name and cost every completion resource.
• Conflating physical and logical Gaussianity. Projection onto a finite code changes the relevant operation, while a non-Gaussian code state alone supplies no universality theorem.
• Hiding resources. Measurements, bits, recovery, frame tracking, rejection, expected repeats, energy, cutoffs, and ancillas belong in the ledger.
• Promoting ideal algebra to protection or evidence. Code preservation and zero modeled leakage do not prove error correction, fault tolerance, threshold behavior, hardware fidelity, or advantage.

For code states, error sets, syndromes, and recovery, use Bosonic Codes and the family pages on Cat Codes, Binomial Codes, and GKP Codes. Direct quadrature computation belongs to Continuous-Variable Quantum Computation. Fault-spread and recovery-performance questions belong to Fault-Tolerant Gates, the Threshold Theorem, and Decoders. Device and dated evidence claims belong to Bosonic Qubits and Claims, Hype, and Evidence Standards.

Exercises

1. Build an Encoding Isometry and Flagged Decoder

Set $W|0_{\mathrm L}\rangle=|0\rangle$ and $W|1_{\mathrm L}\rangle=|2\rangle$. Construct the code projector and flagged decoder. Apply the decoder to $0.9|+_{\mathrm C}\rangle\langle+_{\mathrm C}| +0.1|1\rangle\langle1|$, where $|+_{\mathrm C}\rangle=(|0\rangle+|2\rangle)/\sqrt2$.

Solution

The isometry gives

$$P = |0\rangle\langle0| + |2\rangle\langle2|, \qquad Q = I-P.$$

Define

$$\begin{aligned} \mathcal D_\perp(\rho) =\;& W^\dagger P\rho P W \\ &+ \operatorname{Tr}(Q\rho) |\perp\rangle\langle\perp|. \end{aligned}$$

The code component decodes $|+_{\mathrm C}\rangle$ to $|+\rangle$, whereas $|1\rangle$ lies in the flag subspace. Therefore

$$\mathcal D_\perp(\rho) = 0.9|+\rangle\langle+| + 0.1|\perp\rangle\langle\perp|.$$

Code acceptance is $0.9$ and flag probability is $0.1$. Discarding the flag would falsely turn a trace-preserving decoded channel into an apparently deterministic logical output.

2. Expose the Projection–Composition Trap

Use the encoding $|0_{\mathrm L}\rangle\mapsto|0\rangle$ and $|1_{\mathrm L}\rangle\mapsto|2\rangle$. Let $S$ fix $|0\rangle$ and swap $|1\rangle\leftrightarrow|2\rangle$, with $S|n\rangle=|n\rangle$ for every $n\geq3$. Compare the square of the projected one-step block with the projection of the complete two-step program.

Solution

In the ordered logical basis,

$$W^\dagger S W = \begin{pmatrix} 1&0\\ 0&0 \end{pmatrix},$$

so

$$\left( W^\dagger S W \right)^2 = \begin{pmatrix} 1&0\\ 0&0 \end{pmatrix}.$$

But $S^2=I$, hence

$$W^\dagger S^2W = I_{\mathrm L}.$$

The missing contribution is

$$W^\dagger SQS W = \begin{pmatrix} 0&0\\ 0&1 \end{pmatrix}.$$

The logical-one state leaks fully after the first step and returns after the second. Projected step maps therefore cannot replace the composed physical program.

3. Audit Asymmetric Dual-Rail Loss

Send $|+_{\mathrm L}\rangle=(|10\rangle+|01\rangle)/\sqrt2$ through independent pure-loss channels with $\eta_a=0.9$ and $\eta_b=0.8$. Decode the total-one sector as accepted and vacuum as an orthogonal flag. Find the accepted probability, conditional rail populations, conditional target fidelity, and unconditioned embedded target overlap.

Solution

The unnormalized accepted vector is

$$|\widetilde\psi_{\mathrm{acc}}\rangle = \frac{ \sqrt{\eta_a}|10\rangle + \sqrt{\eta_b}|01\rangle }{ \sqrt2 }.$$

Therefore

$$p_{\mathrm{acc}} = \frac{\eta_a+\eta_b}{2} = 0.85, \qquad p_\perp = 0.15.$$

After normalization, the two rail populations are $9/17$ and $8/17$. The target fidelity is

$$F_{+|\mathrm{acc}} = \frac{ 17+12\sqrt2 }{ 34 } = 0.999134198484622.$$

The unconditioned embedded overlap is

$$F_{+,\mathrm{uncond}} = \frac{ 17+12\sqrt2 }{ 40 } = 0.849264068711929.$$

The accepted probability changes with the logical rail populations when $\eta_a\ne\eta_b$. The normalized accepted state alone is therefore not a deterministic logical channel.

4. Test Code Preservation under a Number Rotation

Encode $|0_{\mathrm L}\rangle=|0\rangle$ and $|1_{\mathrm L}\rangle=|2\rangle$. For $U(\phi)=e^{-i\phi n}$ at $\phi=\pi/2$, find the induced logical action, leakage, output from $|+\rangle$, and input mean photon number.

Solution

Since $U(\pi/2)|0\rangle=|0\rangle$ and $U(\pi/2)|2\rangle=-|2\rangle$,

$$W^\dagger U(\pi/2)W = \operatorname{diag}(1,-1) = Z, \qquad L_U = 0.$$

Thus $|+\rangle\mapsto|-\rangle$, leakage is zero, and the encoded $|+\rangle$ has mean photon number

$$\bar n = \frac12(0+2) = 1.$$

This is one exact native logical $Z$; it does not establish a universal gate set.

5. Normalize an Input-Dependent Accepted Map

Let an accepted and rejected logical instrument have Kraus operators

$$K = \begin{pmatrix} 1&0\\ 0&1/2 \end{pmatrix}, \qquad J = \begin{pmatrix} 0&0\\ 0&\sqrt3/2 \end{pmatrix}.$$

Verify completeness. For $\rho=I/2$, find the accepted probability and normalized state, then test whether normalizing endpoint inputs and averaging gives the same result.

Solution

Direct multiplication gives

$$K^\dagger K + J^\dagger J = I.$$

For $\rho=I/2$,

$$\widetilde\rho_{\mathrm{acc}} = K\rho K^\dagger = \begin{pmatrix} 1/2&0\\ 0&1/8 \end{pmatrix},$$

so

$$p_{\mathrm{acc}} = \frac58, \qquad \rho_{\mathrm{acc}} = \begin{pmatrix} 4/5&0\\ 0&1/5 \end{pmatrix}.$$

The normalized endpoint outputs are $|0\rangle\langle0|$ and $|1\rangle\langle1|$. Their equal average is $I/2$, not the matrix above. Input-dependent normalization is nonlinear, so the subnormalized accepted map and rejected branch must be retained.

6. Bound a Coherent-State Fock Cutoff

For $|\alpha=2\rangle$, consider a dimension-$12$ truncation retaining $n=0,\ldots,11$. Compute the exact omitted probability and compare it with the mean-energy bound.

Solution

The coherent-state number distribution is Poisson with mean $\bar n=|\alpha|^2=4$. Its exact tail is

$$\begin{aligned} \epsilon_{\mathrm{tail}} &= 1 - e^{-4} \sum_{n=0}^{11} \frac{4^n}{n!} \\ &= 0.000915229147270. \end{aligned}$$

The authoritative high-precision value begins $0.000915229147270063013\ldots$; the displayed value is rounded to fifteen decimal places. The mean-energy estimate is only

$$\epsilon_{\mathrm{tail}} \leq \frac{\bar n}{12} = \frac13.$$

Both numbers bound the input-state tail. Neither is a unitary, channel, or logical-gate error certificate.

7. Complete a Native Gate Set and Count Accepted Cost

Explain why arbitrary $R_z(\theta)$ together with $\operatorname{CZ}$ is not universal, and add one declared completion resource. Then audit a known-state program with three sequential layers, each using two sequential native-control slots and one recovery, plus one Hadamard in total. Each recovery uses one binary measurement and one classical bit. One reusable ancilla is prepared before the first recovery and reset or reprepared before the second and third: exactly three ancilla preparation-or-reset slots per attempt. Every recovery accepts independently with probability $4/5$, and a failure consumes the complete attempt.

Solution

Both $R_z(\theta)$ and $\operatorname{CZ}$ are diagonal in the computational basis, so their products remain diagonal and cannot generate arbitrary unitaries. Adding a declared Hadamard gives the completed set

$$\{ R_z(\theta), H, \operatorname{CZ} \}.$$

Only a freshly prepared known input may restart. Three successful recoveries are needed, so

$$p_{\mathrm{acc}} = \left( \frac45 \right)^3 = \frac{64}{125}, \qquad \mathbb E[N_{\mathrm{attempts}}] = \frac{125}{64}.$$

Per attempt, count one fresh known-input preparation, six native controls, one $H$, three recoveries, three binary recovery measurements, three classical bits, one initial ancilla preparation, and two ancilla reset/repreparations. The physical footprint is two storage modes plus one reused ancilla.

Under the declared abstract scheduler, each recovery slot bundles its recovery operation, measurement and bit, and the ancilla preparation or reset/repreparation needed for that recovery. Only under this bundled-slot convention is the depth

$$D_{\mathrm{attempt}} = 3(2+1)+1 = 10.$$

If preparation or reset occupies a separate sequential slot, depth ten is not licensed and the scheduler must recount it. The abstract gate-depth comparator starts after fresh logical-input preparation, while that preparation remains a separately counted resource.

Multiplying consumed operations by the expected attempts gives

$$\mathbb E[N_{\mathrm{native}}] = \frac{375}{32}, \qquad \mathbb E[N_H] = \frac{125}{64},$$ $$\mathbb E[N_{\mathrm{recovery}}] = \mathbb E[N_{\mathrm{meas}}] = \mathbb E[N_{\mathrm{bit}}] = \frac{375}{64}, \qquad \mathbb E[D] = \frac{625}{32}.$$

The expected fresh-input preparation count is $125/64$. The expected total ancilla preparation-or-reset count is $375/64$: $125/64$ initial preparations plus $125/32$ reset/repreparations. The footprint remains three modes and is not multiplied by expected attempts. These are separate resource currencies with no assumed exchange rate. Physical duration, preparation and reset latency or infidelity, energy, pulse-level decomposition, calibration, noise, fault-tolerant overhead, and a hardware comparator lie outside this abstract comparator rather than being silently priced at zero. Local retry would define a different protocol. This ledger licenses neither fault tolerance nor advantage.

8. Complete a Ten-Field Single-Rail Phase Program Record

Complete the full record for a fixed $|+\rangle$ single-rail input, the ideal number rotation $U=e^{-i\pi n/2}$, and a terminal logical-$Y$ measurement.

Solution

1. Computational task, input family, and licensed claim — On the declared logical input $|+\rangle$, implement a single-rail phase program and predict one final logical-$Y$ outcome under the promise that the measured state remains in the code, $P\rho P=\rho$. The licensed claim is exact only for this ideal one-mode program; the two-outcome instrument is not asserted to be complete on arbitrary full-Fock-space inputs, and no hardware or fault-tolerance claim is made.

2. Mode register, encoding isometry, code projector, basis, and energy convention — Use one mode with $W|0\rangle=|0\rangle$, $W|1\rangle=|1\rangle$, $P=|0\rangle\langle0|+|1\rangle\langle1|$, and number convention $n=0,1,\ldots$. The encoded input has mean photon number $1/2$.

3. Physical controls, noise, measurements, domains, and promises — Use the ideal number rotation

   $$U = e^{-i\pi n/2}.$$

   Its action is evaluated on the invariant $n=0,1$ code. Define the physical code-space $Y$ eigenstates, projectors, and Lüders Kraus operators by

   $$|y_\pm^{\mathrm C}\rangle := W|y_\pm\rangle = \frac{ |0\rangle\pm i|1\rangle }{ \sqrt2 },$$ $$\Pi_\pm := |y_\pm^{\mathrm C}\rangle \langle y_\pm^{\mathrm C}|, \qquad M_\pm := \Pi_\pm.$$

   The ideal instrument is $\mathcal I_\pm(\rho)=M_\pm\rho M_\pm^\dagger$. Because $\Pi_++\Pi_-=P$, it is a complete two-outcome instrument only on the promised code. For unrestricted physical inputs, a complete extension must add $M_\perp=Q$ and its flag outcome. No noise or physical implementation is licensed.

4. Preparation, ancillas, recovery, and logical-frame convention — Prepare $W|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$. There is no ancilla, physical recovery, or decoded frame update.

5. Program order, classical record, adaptivity, and accepted branches — Apply the single phase operation, then the declared $\{\mathcal I_+,\mathcal I_-\}$ instrument. Record its one-bit outcome. The code promise and code-preserving phase make the $Q$ flag impossible in this exercise. There is no adaptive branch, rejection, or retry.

6. Induced logical map, leakage, rejection, and normalization — The code is preserved and

   $$W^\dagger U W = \operatorname{diag}(1,-i) = S^\dagger.$$

   Leakage and rejection are zero, so no conditional normalization is used. The terminal logical branch maps are $\rho\mapsto\pi_\pm S^\dagger\rho S\pi_\pm$, where $\pi_\pm=(I\pm Y)/2$.

7. Output, decoder, target, and error metric — The premeasurement output is

   $$|y_-\rangle = \frac{ |0\rangle-i|1\rangle }{ \sqrt2 },$$

   so the ideal logical-$Y$ instrument returns eigenvalue $-1$ with probability $1$ and leaves the same conditional postmeasurement state. The target classical law is the point mass at $-1$, with total-variation error zero; exact premeasurement state equality is checked separately.

8. Resources, fault assumptions, comparator, and owner boundary — Count one mode, mean input number $1/2$, one phase operation at phase-control depth one, one terminal logical-$Y$ instrument, and one classical bit under the code promise. Ancillas, recovery, feedforward, rejected attempts, and repeats are zero. For an unrestricted input, the additional $Q$ outcome and a record capable of representing it would belong to a different ledger. A physical implementation, time, precision, calibration, noise, and fault-tolerant overhead are N/A because only an abstract exact operation and code-promised instrument are licensed.

9. Verification data, truncation control, uncertainty, and reproducibility — Verify the two code phases, zero leakage, $\Pi_++\Pi_-=P$, $M_+^\dagger M_++M_-^\dagger M_-=P$, $Y|y_-\rangle=-|y_-\rangle$, and unit probability exactly and in an independent binary64 evaluator to $10^{-12}$. No cutoff is required on the invariant code. Experimental uncertainty is N/A.

10. Conclusion, stopping point, and canonical handoff — The record verifies one exact fixed-input single-rail phase program and its resource ledger. It proves no error correction, protection, universality, fault tolerance, hardware performance, or advantage; those claims belong to Bosonic Codes, Universal Gate Sets, fault-tolerance, hardware, and evidence owners.

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