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Single-shot GHZ characterization with connectivity-aware fanout constructions

Giancarlo Gatti

TL;DR

Problem addressed: enabling ancilla-free, low-depth fan-out gates for large GHZ states under realistic device connectivities. Approach: convert a depth-$L$ GHZ-constructing CNOT block into a depth-$2L-1$ fan-out, yielding $2\log_2(n)-1$ in fully connected devices and $O(n^{1/2})$ under heavy-hex connectivity; demonstration on ibm_fez with a 156-qubit system gives fan-out depth $33$ and enables single-shot measurement of $n$-body Pauli contexts at that depth. Contributions: generalized fan-out equivalence with explicit construction, numerical validation up to $n=16$, and a practical architecture-level demonstration for GHZ-state characterization. Significance: provides an architecture-aware, ancilla-free pathway to scalable GHZ-state manipulation and rapid context-based state tomography on current quantum hardware.

Abstract

We propose a practical recipe to transform any depth-$L$ block of CNOTs that prepares $n$-qubit GHZ states into an $n$-qubit fanout gate (multitarget-CNOT) of depth $2L-1$, without the need for ancilla qubits. Considering known logarithmic-depth circuits to prepare GHZ-states, this allows us to construct an $n$-qubit fanout gate with depth $2\log_2(n)-1$, reproducing previous ancillaless constructions. We employ our recipe to construct $n$-qubit fanout gates under heavy-hex connectivity restrictions, obtaining a depth of $O(n^{1/2})$, again reproducing previous complexity theory constructions. Using this recipe on the \textit{ibm\_fez} architecture yields a $156$-qubit fanout construction with depth $33$. Additionally, we show how to employ these $n$-qubit fanout constructions to measure complete sets of commuting observables from the $n$-body Pauli group with the same depth, allowing for efficient single-shot characterization of any GHZ-like state in a given known basis, e.g. fully characterizing a single copy of a $156$-qubit GHZ state using circuit depth $33$ in $\textit{ibm\_fez}$ (its preparation requires an additional depth of $17$).

Single-shot GHZ characterization with connectivity-aware fanout constructions

TL;DR

Problem addressed: enabling ancilla-free, low-depth fan-out gates for large GHZ states under realistic device connectivities. Approach: convert a depth- GHZ-constructing CNOT block into a depth- fan-out, yielding in fully connected devices and under heavy-hex connectivity; demonstration on ibm_fez with a 156-qubit system gives fan-out depth and enables single-shot measurement of -body Pauli contexts at that depth. Contributions: generalized fan-out equivalence with explicit construction, numerical validation up to , and a practical architecture-level demonstration for GHZ-state characterization. Significance: provides an architecture-aware, ancilla-free pathway to scalable GHZ-state manipulation and rapid context-based state tomography on current quantum hardware.

Abstract

We propose a practical recipe to transform any depth- block of CNOTs that prepares -qubit GHZ states into an -qubit fanout gate (multitarget-CNOT) of depth , without the need for ancilla qubits. Considering known logarithmic-depth circuits to prepare GHZ-states, this allows us to construct an -qubit fanout gate with depth , reproducing previous ancillaless constructions. We employ our recipe to construct -qubit fanout gates under heavy-hex connectivity restrictions, obtaining a depth of , again reproducing previous complexity theory constructions. Using this recipe on the \textit{ibm\_fez} architecture yields a -qubit fanout construction with depth . Additionally, we show how to employ these -qubit fanout constructions to measure complete sets of commuting observables from the -body Pauli group with the same depth, allowing for efficient single-shot characterization of any GHZ-like state in a given known basis, e.g. fully characterizing a single copy of a -qubit GHZ state using circuit depth in (its preparation requires an additional depth of ).
Paper Structure (5 sections, 1 theorem, 20 equations, 3 figures, 1 table)

This paper contains 5 sections, 1 theorem, 20 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions
  3. Generalized fan-out equivalence
  4. GHZ-preparation and context measurement in $156$-qubit heavy-hex topology
  5. Conclusions and Outlook

Key Result

Proposition 1

Consider a list $\mathcal{U}_\text{GHZ}$ of $\text{CX}(i,j)$ gates such that their application from left to right maps the $\ket{+}_h\ket{0}_{t_1}\ket{0}_{t_2}\text{...}\ket{0}_{t_{n-1}}$ state into the $n$-qubit GHZ state $\ket{\text{GHZ}_0}=\tfrac{1}{\sqrt{2}}(\ket{0}^{\otimes n}+\ket{1}^{\otimes is equivalent under application to a fan-out gate $\text{CX}(h\rightarrow t_1,t_2,\text{...},t_{n-1

Figures (3)

  • Figure 1: Fan-out gate equivalence. A multi-target CNOT, or fan-out gate can be constructed by concatenating two CNOT ladders, thus having an "open" and "closed" equivalence.
  • Figure 2: Fan-out gate implementation examples. Optimal logarithmic-depth fan-out gate implementations for $n=4,8,16,32$ qubits, with respective depth $D=3,5,7,9$. To produce this construction, we employ a GHZ-generating block of CNOTs $\mathcal{U}_\text{GHZ}$ and its inverse with the CNOTs in red removed.
  • Figure 3: Connectivity and layer map for GHZ-state preparation in ibm_fez. To prepare a $156$-qubit GHZ state, a Hadamard gate is applied on qubit number $69$ followed by $155$ CNOTs, which can be grouped into $17$ layers. In this map, the qubits are indexed between $0$ and $155$, and all qubits excepting qubit $69$ are also accompanied by a number between $1$ and $17$, which indicates the layer in which they are targeted by a CNOT, with one of their entangled neighbours being the control. Additionally, we color-code groups of consecutive layers for easier visualization.

Theorems & Definitions (5)

  • Definition 2.1: $n$-body Pauli observables
  • Definition 2.2: $n$-body Pauli contexts
  • Definition 2.3: $n$-qubit GHZ-class states
  • Proposition 1
  • proof