---
title: Probabilistic Quantum Circuits with Fallback (PQF)
url: https://www.emergentmind.com/topics/probabilistic-quantum-circuits-with-fallback-pqf
type: topic
---

# Probabilistic Quantum Circuits with Fallback (PQF)

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 [1409.3552][2203.10064].

## 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 $F_j$, where $F_0=I$ and $F_k=G$ is the absorbing (target) state. Each round applies a multi-qubit unitary followed by ancilla measurement, with transition probabilities reflecting primary circuit success ($p_j$) or failure ($1-p_j$). In contrast to RUS, PQF always halts after at most $k+1$ rounds, with each round potentially using a distinct primary circuit [1409.3552].

## 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 $k$ by design, permitting stage-dependent circuit variation, and always triggers fallback at the prescribed round, leading to robust predictability and potentially lower resource variance [1409.3552].

## 3. PQF Synthesis Workflow for Single-Qubit Rotations

The synthesis of a PQF circuit for an arbitrary single-qubit rotation $R_z(\theta)$ involves four main algorithmic stages:

1. **Cyclotomic/Rational Approximation**: Compute $z \in \mathbb{Z}[\omega]$ such that $|z^*/z - e^{i\theta}| < \varepsilon$, with $|z| = O(\varepsilon^{-1/d})$ and $d=\varphi(m)$, using PSLQ or related algorithms.

2. **Probability Enhancement**: Search for $r\in \mathbb{Z}[p]$ such that $|z'|^2 / b^{2L} > 1 - O(1/L)$ after scaling $z' = r z$ and $L = \lceil \log_b |z'| \rceil$, for gate-set-specific $b$ (e.g., $b=2$ for Clifford+$T$). Solve the norm equation $|y|^2 = b^{2L} - |z'|^2$ efficiently for $y \in \mathbb{Z}[\omega]$.

3. **Primary Probabilistic Step**: Construct
   \[
   V = \frac{1}{b^L}
   \begin{bmatrix}
   z' & -y^* \\
   y & z'^*
   \end{bmatrix}
   \]
   and realize the two-qubit gate $U = \operatorname{CNOT}\cdot (I\otimes V)\cdot\operatorname{CNOT}$.

4. **Measurement and Fallback**: Apply $U$ to $|\psi\rangle\otimes|0\rangle$, measure ancilla; on success, the target rotation is achieved. On failure, the data qubit is in $R_z(\theta')|\psi\rangle$ for some $\theta' \neq \theta$, and a deterministic fallback $F$ is computed and applied to complete the rotation to the desired accuracy [1409.3552].

## 4. Cost Analysis and Gate Count Asymptotics

Let $C_p$ be the cost per probabilistic round and $C_F$ the fallback cost. For uniform success probability $p$ per round, the expected cost and variance over $k$ rounds are:
\[
\mathbb{E}[\text{Cost}] = \frac{C_p}{p} + O(q^k),\quad \text{Var}[\text{Cost}] = C_p^2 \frac{q}{p^2} + O(q^k),\quad \text{where } q=1-p.
\]

For single-qubit rotations using PQF, the expected non-Clifford gate cost is
\[
\mathbb{E}[\text{Cost}] = \log_b(1/\varepsilon) + O(\log\log(1/\varepsilon)),
\]
with $b$ the expansion factor associated with the gate set (Clifford+$T$: $b=2$; Clifford+$V$: $b=5$; Clifford+$\pi/12$: $b=4$). Numeric fits yield:

| Gate Set                 | $b$ | Asymptotic Expected Non-Clifford Count                      |
|--------------------------|-----|------------------------------------------------------------|
| Clifford+$T$             | 2   | $\log_2(1/\varepsilon) + 4\log_2\log_2(1/\varepsilon) + 1.19$ |
| Clifford+$V$             | 5   | $\log_5(1/\varepsilon) + 0.95\log_5\log_5(1/\varepsilon) + 7.26$ |
| Clifford+$\pi/12$        | 4   | $\frac{1}{2}\log_2(1/\varepsilon) + 2\log_2\log_2(1/\varepsilon) + 3.48$ |

This scaling matches the leading asymptotics of RUS while ensuring finite-mean and finite-variance termination [1409.3552].

## 5. Fallback Paradigm and Mixed-Channel Optimization

PQF protocols further benefit from the "fallback circuit paradigm" and mixed-channel construction. The standard fallback channel $\Lambda_\text{fallback}$ 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:
\[
\|Z_\theta - \Lambda_\text{fallback}\|_\diamond \leq \varepsilon_1 + (1-p)\varepsilon_2,
\]
where $p$ is the projective step's success probability and $\varepsilon_1$, $\varepsilon_2$ 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 $\Lambda_\text{mix} = p \Lambda_1 + (1-p)\Lambda_2$, the diamond-norm error scales as $O(\varepsilon^2)$, halving the required non-Clifford gate count:
- Standard fallback: area $\sim\varepsilon$ in parameter space $\Rightarrow N \sim \log_2(1/\varepsilon)$
- Mixed fallback: area $\sim\sqrt{\varepsilon}$ $\Rightarrow N \sim 0.5 \log_2(1/\varepsilon)$

Numerical results confirm that, for Clifford+$T$, the mixed-fallback protocol achieves $0.53\log_2(1/\varepsilon) + 4.86$ (T-count); for Clifford+$\sqrt{T}$, $0.56\log_2(1/\varepsilon) + 5.3$ (T-count), representing an approximate factor-of-two improvement over basic fallback [2203.10064].

## 6. Reduction to Magnitude Approximation and Algorithmic Synthesis

For general single-qubit $U\in SU(2)$, magnitude-only approximations—reducing to matching the absolute value of a key matrix element—enable further gate savings. The protocol reduces $U$ to an Euler product $Z_{\varphi_1} X_\theta Z_{\varphi_2}$; only the $X_\theta$ component is approximated via magnitude, up to unknown $Z$ phases, resulting in total cost $7/9$ that of full diagonalization.

The synthesis algorithm over gate set $G$ 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 $G$.
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 [2203.10064].

## 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+$T$ gate sets, the mixed-fallback protocol achieves mean T-count scaling with slope ≈0.53; for Clifford+$\sqrt{T}$, ≈0.56. Protocols exhibit mean and worst-case costs tightly controlled by per-round success probabilities ($q\approx0.99$), and cost amplification can be mitigated for parallel tasks by increasing primary circuit success [2203.10064].

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.

Source: https://www.emergentmind.com/topics/probabilistic-quantum-circuits-with-fallback-pqf