Probabilistic Quantum Circuits with Fallback (PQF)

• PQF is a synthesis protocol that approximates arbitrary single-qubit rotations using a combination of measurement-induced probabilistic circuits and a deterministic fallback to guarantee termination.
• It employs a discrete finite Markov chain model with stage-dependent circuit variations and mixing strategies to control cost and reduce non-Clifford gate counts.
• Cost analysis shows that PQF achieves predictable mean and variance in resource usage, often halving the required gate count compared to traditional RUS circuits.

Probabilistic Quantum Circuits with Fallback (PQF) are a class of synthesis protocols for approximating arbitrary single-qubit unitaries using a finite universal gate set, in which the circuit employs measurement-induced probabilistic subcircuits and guarantees finite expected resource cost by terminating after a bounded number of rounds with a deterministic fallback. PQF generalizes and improves upon the Repeat-Until-Success (RUS) paradigm by leveraging finite Markov chains for circuit progress, explicit cost control, and by incorporating advanced techniques such as mixing and magnitude-approximation to further reduce gate complexity (Bocharov et al., 2014, Kliuchnikov et al., 2022).

1. Formal Structure and Markov Model of PQF Circuits

A PQF protocol implements a target unitary $G$ within diamond-norm accuracy $\varepsilon$ using at most $k$ probabilistic "primary" circuits, each terminating upon success, with a deterministic "fallback" subcircuit in the event of sequential failure. The protocol is recursively defined as: $$\begin{align*} \mathrm{PQF}(G, \varepsilon, 0) &= F(G, \varepsilon), \ \mathrm{PQF}(G, \varepsilon, k) &= P(G, \varepsilon) \oplus_\mathrm{BC} \mathrm{PQF}(G G_1, \varepsilon, k-1), \end{align*}$$ where $P(G,\varepsilon)$ is a probabilistic subcircuit for $G$ (success probability $p>0$), $G_1 \neq G$ is the accumulated residual unitary in case of failure, $F(G,\varepsilon)$ is the deterministic fallback, and $\oplus_\mathrm{BC}$ denotes classical control on the measurement outcome.

The execution of PQF is equivalent to a walk on a discrete, finite Markov chain: each node represents an accumulated operation $\varepsilon$0, where $\varepsilon$1 and $\varepsilon$2 is the absorbing (target) state. Each round applies a multi-qubit unitary followed by ancilla measurement, with transition probabilities reflecting primary circuit success ($\varepsilon$3) or failure ($\varepsilon$4). In contrast to RUS, PQF always halts after at most $\varepsilon$5 rounds, with each round potentially using a distinct primary circuit (Bocharov et al., 2014).

2. Comparison to Repeat-Until-Success (RUS) Circuits

Both PQF and RUS leverage probabilistic subcircuits and measurement but differ fundamentally in termination guarantees and flexibility. RUS protocols iterate a fixed probabilistic circuit and correction until success, defining an infinite-length Markov chain (absorbing on success). Expected cost can be unpredictable in rare pathological instances. PQF strictly bounds the total rounds by $\varepsilon$6 by design, permitting stage-dependent circuit variation, and always triggers fallback at the prescribed round, leading to robust predictability and potentially lower resource variance (Bocharov et al., 2014).

3. PQF Synthesis Workflow for Single-Qubit Rotations

The synthesis of a PQF circuit for an arbitrary single-qubit rotation $\varepsilon$7 involves four main algorithmic stages:

1. Cyclotomic/Rational Approximation: Compute $\varepsilon$8 such that $\varepsilon$9, with $k$0 and $k$1, using PSLQ or related algorithms.
2. Probability Enhancement: Search for $k$2 such that $k$3 after scaling $k$4 and $k$5, for gate-set-specific $k$6 (e.g., $k$7 for Clifford+$k$8). Solve the norm equation $k$9 efficiently for $$\begin{align*} \mathrm{PQF}(G, \varepsilon, 0) &= F(G, \varepsilon), \ \mathrm{PQF}(G, \varepsilon, k) &= P(G, \varepsilon) \oplus_\mathrm{BC} \mathrm{PQF}(G G_1, \varepsilon, k-1), \end{align*}$$0.
3. Primary Probabilistic Step: Construct

$$\begin{align*} \mathrm{PQF}(G, \varepsilon, 0) &= F(G, \varepsilon), \ \mathrm{PQF}(G, \varepsilon, k) &= P(G, \varepsilon) \oplus_\mathrm{BC} \mathrm{PQF}(G G_1, \varepsilon, k-1), \end{align*}$$1

and realize the two-qubit gate $$\begin{align*} \mathrm{PQF}(G, \varepsilon, 0) &= F(G, \varepsilon), \ \mathrm{PQF}(G, \varepsilon, k) &= P(G, \varepsilon) \oplus_\mathrm{BC} \mathrm{PQF}(G G_1, \varepsilon, k-1), \end{align*}$$2.

1. Measurement and Fallback: Apply $$\begin{align*} \mathrm{PQF}(G, \varepsilon, 0) &= F(G, \varepsilon), \ \mathrm{PQF}(G, \varepsilon, k) &= P(G, \varepsilon) \oplus_\mathrm{BC} \mathrm{PQF}(G G_1, \varepsilon, k-1), \end{align*}$$3 to $$\begin{align*} \mathrm{PQF}(G, \varepsilon, 0) &= F(G, \varepsilon), \ \mathrm{PQF}(G, \varepsilon, k) &= P(G, \varepsilon) \oplus_\mathrm{BC} \mathrm{PQF}(G G_1, \varepsilon, k-1), \end{align*}$$4, measure ancilla; on success, the target rotation is achieved. On failure, the data qubit is in $$\begin{align*} \mathrm{PQF}(G, \varepsilon, 0) &= F(G, \varepsilon), \ \mathrm{PQF}(G, \varepsilon, k) &= P(G, \varepsilon) \oplus_\mathrm{BC} \mathrm{PQF}(G G_1, \varepsilon, k-1), \end{align*}$$5 for some $$\begin{align*} \mathrm{PQF}(G, \varepsilon, 0) &= F(G, \varepsilon), \ \mathrm{PQF}(G, \varepsilon, k) &= P(G, \varepsilon) \oplus_\mathrm{BC} \mathrm{PQF}(G G_1, \varepsilon, k-1), \end{align*}$$6, and a deterministic fallback $$\begin{align*} \mathrm{PQF}(G, \varepsilon, 0) &= F(G, \varepsilon), \ \mathrm{PQF}(G, \varepsilon, k) &= P(G, \varepsilon) \oplus_\mathrm{BC} \mathrm{PQF}(G G_1, \varepsilon, k-1), \end{align*}$$7 is computed and applied to complete the rotation to the desired accuracy (Bocharov et al., 2014).

4. Cost Analysis and Gate Count Asymptotics

Let $$\begin{align*} \mathrm{PQF}(G, \varepsilon, 0) &= F(G, \varepsilon), \ \mathrm{PQF}(G, \varepsilon, k) &= P(G, \varepsilon) \oplus_\mathrm{BC} \mathrm{PQF}(G G_1, \varepsilon, k-1), \end{align*}$$8 be the cost per probabilistic round and $$\begin{align*} \mathrm{PQF}(G, \varepsilon, 0) &= F(G, \varepsilon), \ \mathrm{PQF}(G, \varepsilon, k) &= P(G, \varepsilon) \oplus_\mathrm{BC} \mathrm{PQF}(G G_1, \varepsilon, k-1), \end{align*}$$9 the fallback cost. For uniform success probability $P(G,\varepsilon)$0 per round, the expected cost and variance over $P(G,\varepsilon)$1 rounds are: $P(G,\varepsilon)$2

For single-qubit rotations using PQF, the expected non-Clifford gate cost is

$P(G,\varepsilon)$3

with $P(G,\varepsilon)$4 the expansion factor associated with the gate set (Clifford+$P(G,\varepsilon)$5: $P(G,\varepsilon)$6; Clifford+$P(G,\varepsilon)$7: $P(G,\varepsilon)$8; Clifford+$P(G,\varepsilon)$9: $G$0). Numeric fits yield:

Gate Set	$G$1	Asymptotic Expected Non-Clifford Count
Clifford+$G$2	2	$G$3
Clifford+$G$4	5	$G$5
Clifford+$G$6	4	$G$7

This scaling matches the leading asymptotics of RUS while ensuring finite-mean and finite-variance termination (Bocharov et al., 2014).

5. Fallback Paradigm and Mixed-Channel Optimization

PQF protocols further benefit from the "fallback circuit paradigm" and mixed-channel construction. The standard fallback channel $G$8 consists of a probabilistic step (projective rotation), yielding either the target or a nearby unitary, followed by deterministic correction in case of failure. Errors compose additively: $G$9 where $p>0$0 is the projective step's success probability and $p>0$1, $p>0$2 the respective step errors.

A significant improvement is achieved by mixing two such fallback protocols (one under-rotating, one over-rotating) with optimized probabilities. For the composite channel $p>0$3, the diamond-norm error scales as $p>0$4, halving the required non-Clifford gate count:

• Standard fallback: area $p>0$5 in parameter space $p>0$6
• Mixed fallback: area $p>0$7 $p>0$8

Numerical results confirm that, for Clifford+$p>0$9, the mixed-fallback protocol achieves $G_1 \neq G$0 (T-count); for Clifford+$G_1 \neq G$1, $G_1 \neq G$2 (T-count), representing an approximate factor-of-two improvement over basic fallback (Kliuchnikov et al., 2022).

6. Reduction to Magnitude Approximation and Algorithmic Synthesis

For general single-qubit $G_1 \neq G$3, magnitude-only approximations—reducing to matching the absolute value of a key matrix element—enable further gate savings. The protocol reduces $G_1 \neq G$4 to an Euler product $G_1 \neq G$5; only the $G_1 \neq G$6 component is approximated via magnitude, up to unknown $G_1 \neq G$7 phases, resulting in total cost $G_1 \neq G$8 that of full diagonalization.

The synthesis algorithm over gate set $G_1 \neq G$9 proceeds as:

1. Enumerate candidate integer points (via, e.g., Lenstra's algorithm) satisfying the angle, magnitude, and probabilistic constraints in the quaternion order associated with $F(G,\varepsilon)$0.
2. For each candidate, solve the associated relative-norm equation to guarantee implementability.
3. If solutions for both under- and over-rotations are found, mix with optimized probability to achieve error reduction.
4. Synthesize the corresponding gate sequences using known quaternion-based algorithms (Kliuchnikov et al., 2022).

7. Numerical Benchmarks and Applicability to Fault-Tolerant Quantum Computation

Benchmarks over datasets of random and Fourier angles confirm the asymptotic scaling and variance bounds for PQF circuits. For Clifford+$F(G,\varepsilon)$1 gate sets, the mixed-fallback protocol achieves mean T-count scaling with slope ≈0.53; for Clifford+$F(G,\varepsilon)$2, ≈0.56. Protocols exhibit mean and worst-case costs tightly controlled by per-round success probabilities ($F(G,\varepsilon)$3), and cost amplification can be mitigated for parallel tasks by increasing primary circuit success (Kliuchnikov et al., 2022).

PQF protocols, particularly when leveraging magnitude-only decompositions and mixed fallback constructions, provide the lowest known non-Clifford resource requirements for single-qubit synthesis under diamond-norm accuracy, and are thus foundational for efficient logical gate synthesis in fault-tolerant quantum computing systems.