Quantum advantage from effective $200$-qubit holographic random circuit sampling

TL;DR

The paper introduces holographic random circuit sampling (HRCS), a framework that couples a small system of qubits with a larger bath and uses mid-circuit measurements to translate circuit depth into additional effective qubits, exponentially expanding sampling complexity beyond conventional random circuit sampling. It provides rigorous expressions showing how the joint spatiotemporal distribution preserves anticoncentration up to exponential time scales and derives a bound on the growth of the effective sampling dimension $N_{ m eff}(t)$, along with generalizations to higher-order statistics. Experimentally, HRCS demonstrates substantial scalability on IBM hardware: with 10 physical qubits, effective sampling up to 200 qubits is achieved (via $N_A=10$, $N_B=10$, and $t=19$ steps) and XEB fidelities reach significant values; larger patches further illustrate the method’s potential toward quantum advantage with modest hardware. The work establishes a new route to scalable quantum advantage by exploiting both spatial and temporal quantum resources, offering a quantitative benchmark framework (XEB) and predictive noisy-theory for practical near-term devices.

Abstract

Quantum computers hold the promise of outperforming classical computers in solving certain problems. While large-scale quantum algorithms will require fault-tolerant devices, near-term demonstrations of quantum advantage on existing devices can provide important milestones. Random circuit sampling has emerged as a leading candidate for such demonstrations. However, existing implementations often underutilize circuit depth, limiting the achievable advantage. We introduce a holographic random circuit sampling algorithm that substantially increases the sampling complexity by leveraging repeated interactions and mid-circuit measurements. This approach scales the effective sampling dimension with the circuit depth, ultimately leading to an exponential growth in sampling complexity. With merely 20 physical qubits on IBM quantum devices, we experimentally demonstrate the effective sampling of up to 200 qubits, with a cross-entropy benchmark fidelity of $0.0593$, establishing a new route to scalable quantum advantage through the combined use of spatial and temporal quantum resources.

Figures (11)

• Figure 1: A schematic for quantum circuit sampling methods and main results. In (a) random circuit sampling (RCS), a random unitary circuit is applied on a trivial initial state and one performs computational basis measurements on the output state. (b) In holographic random circuit sampling (HRCS), computational basis measurements are performed on the bath $B$ in each step following a unitary circuit, and on the system $A$ in the end. The sampling task is targeted at the joint distribution of temporal mid-circuit measurements ${\bm z}(t)$ and the final state measurements $x(t)$. In (c), we summarize the anticoncentration (AC) of sampling in HRCS in the asymptotic limit of $N_A \to \infty$ for increasing effective system size in HRCS. While RCS is a single operating point (magenta diamond), HRCS allows the increase of effective system size (blue curve). The inset shows the decay of collision probability (in logarithmic scale) with increasing number of temporal steps.
• Figure 2: Theory and Classical-verifiable benchmark of HRCS. (a) Ensemble-averaged collision probability (CP) $Z(t)$ for HRCS versus temporal steps in a system of $N_A=6$, $N_B=1, 2$ (blue and orange circles) qubits. Colored solid lines represent theoretical result of $Z_{\rm HRCS}(t)$ of Eq. \ref{['eq:HRCS_CP_spt']} in Theorem \ref{['HRCS_CP_spt']}. Dark-colored dashed lines are CP for Haar random states $Z_{\rm H}(N_{\rm eff})$ in Eq. \ref{['ZH_overview']} with effective number of qubits $N_{\rm eff}={N_A+tN_B}$. Inset (a1) shows the growth of critical temporal steps $\tau$ versus system size $N_A$, with $\epsilon=1$. Blue and orange crosses represent numerical solutions of $Z_{\rm HRCS}(t) = (1+\epsilon)Z_{\rm H}(N_{\rm eff})$ for $N_B=2$ and $N_B=6$ separately. The black line is Eq. \ref{['eq:max_steps_overview']}. (b1-b4) Ensemble-averaged probability density function of the joint sampling distribution $P[{\bm z}(t), x(t)]$ in HRCS of $N_A=6, N_B=4$ qubits from $t=1$ to $4$. Red dashed line is the Porter-Thomas distribution corresponding to Haar case. The ensemble average is over $50$ HRCS instances with corresponding error bars. (d) Experimental verification of cross-entropy benchmark fidelity ${\cal F}_{\rm XEB}(t)$ in HRCS. Orange circles and dashed lines show the noiseless simulation of HRCS and theory. Red filled circles show HRCS experimental results on IBMQ Torino ($t=1,2,4,6,8,10,13,16$) and red dashed line is the noisy theory prediction of Eq. \ref{['eq:noisy_xeb']}. Each circuit in HRCS experiment is an $8$-layer hardware-efficient ansatz of $N_A=N_B=5$ qubits. We perform $10^6$ shots for sampling each circuit instance and each circle is an average over $10$ instances. Dark grey circles in inset indicate the used qubits in experiments for $t\le 10$. Blue up and green down triangles are known recent experiments from Google (2019 arute2019quantum and 2024 morvan2024phase) and USTC (2021 wu2021strong, 2022 zhu2022quantum and 2025 gao2025establishing) for comparison. Data points for benchmarking are highlighted by color filling with ($N_{\rm eff}$, ${\cal F}_{\rm XEB}(N_{\rm eff})$) specified.
• Figure 3: Large-scale benchmark of HRCS. Experimental verification of cross-entropy benchmark fidelity ${\cal F}_{\rm XEB}(t)$ in large-scale HRCS. Orange dashed line show the noiseless theory of HRCS. Red filled circuits show experimental results of HRCS on IBMQ Torino $(t=1, 2, 3, 5, 7, 9, 12, 15, 19)$ and red dashed line is the noisy theory prediction. The experiments utilizes 20 qubits in two patches. We perform $10^6$ shots for sampling each circuit instance and each circle is an average over $10$ circuit instances. Dark grey dots in inset show the used qubits on IBMQ Torino for one circuit instance at $t=19$. Blue and green triangles represent the RCS results with largest size in known recent experiments from Google and USTC arute2019quantummorvan2024phasewu2021strongzhu2022quantumgao2025establishing.
• Figure 4: Statistical measure for HRCS (a) Ensemble-averaged power sums (PS) $Z^{(K)}(t)$ for HRCS versus temporal steps in a system of $N_A = 6, N_B=1$ qubits with $K=3, 4$ (blue and orange circles). Colored solid lines represent theoretical results of Eq. \ref{['eq:HRCS_PS_spt']} in Theorem \ref{['HRCS_PS_spt']}. Dark-colored dashed lines are PS for Haar random states $Z_{\rm H}^{(K)}(N_{\rm eff})$ with $N_{\rm eff}=N_A + t N_B$. (b) Ensemble-averaged total variation distance (TVD) versus temporal steps $t$ in a system of $N_A=4, N_B=1, 2$ (blue and orange circles). Colored and black dashed lines represent the first and second upper bounds in Ineq. \ref{['TVD_UB']}. In both subplots, we average over $50$ HRCS circuit instances and in (b) we also average over $100$ Haar random states.
• Figure 5: Schematic of the equivalent expansion of bath systems in HRCS for different sampling. Here we show an example of $t=3$. In (d), we show the schematic for temporal sampling at the second step, with distribution $p[z_2]$.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Lemma 3
• Lemma 4
• Lemma 5
• Corollary 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10