Quantum Error Mitigation as a Universal Error Reduction Technique: Applications from the NISQ to the Fault-Tolerant Quantum Computing Eras

In the early years of fault-tolerant quantum computing (FTQC), it is expected that the available code distance and the number of magic states will be restricted due to the limited scalability of quantum devices and the insufficient computational power of classical decoding units. Here, we integrate quantum error correction and quantum error mitigation into an efficient FTQC architecture that effectively increases the code distance and $T$-gate count at the cost of constant sampling overheads in a wide range of quantum computing regimes. For example, while we need ${10}^4$ to ${10}^{10}$ logical operations for demonstrating quantum advantages from optimistic and pessimistic points of view, we show that we can reduce the required number of physical qubits by 80% and 45% in each regime. From another perspective, when the achievable code distance is up to about 11, our scheme allows executing ${10}^3$ times more logical operations. This scheme will dramatically alleviate the required computational overheads and hasten the arrival of the FTQC era.

Popular Summary

A noisy intermediate-scale quantum (NISQ) device with over 50 qubits has demonstrated quantum supremacy, i.e., that quantum computers can outperform classical computers in solving a specific problem. The next milestone is to use such quantum devices to solve practical problems, e.g., in quantum chemistry and machine learning. Since quantum computers in the near future are expected to be very noisy, and the number of qubits is restricted, quantum error mitigation (QEM) for obtaining correct results without significant hardware overheads will be indispensable to make the best of the computational power of NISQ computers.

When quantum technologies become mature, scientists will try to realize universal fault-tolerant quantum computers that perform calculations while correcting errors by detecting errors and using adaptive feedback with qubit overheads, which is called quantum error correction (QEC). Here, we surely face a similar problem in the intermediate fault-tolerant quantum computing (FTQC) era; the size and resources of quantum computers are insufficient to achieve the required accuracy, and accordingly, we obtain noisy results. Thus, we again need to compensate for computation errors without increasing hardware overheads. To overcome this problem, we propose a new FTQC architecture where QEM and QEC contribute equally. While it is necessary to apply QEC recovery operations with the aid of classical software for implementing quantum algorithms fault tolerantly, we show that our framework can perform QEM and QEC simply with classical postprocessing. Therefore, our method is compatible with FTQC and will improve computation accuracy dramatically.

Figure 1

Schematic picture representing the transitional period from the classically tractable FTQC regime towards the realization of the long-term FTQC. We are now in the classically tractable FTQC era due to the lack of physical qubits [11, 12]. In the figure, the purple line indicates the hardware requirement for performing classically intractable tasks with a realistic time whereas the blue line corresponds to the requirement for demonstrating quantum advantages with conventional long-term quantum algorithms. To estimate these lines, we refer to the quantum supremacy experiments [13] and the existing state-of-the-art resource estimation [9, 10, 14]. The early FTQC regime is defined as a region between these lines. In the main text, we assume that the number of error events during FTQC $N_e$ is required to be smaller than ${10}^{-3}$, which is shown as the dotted black line. Our technique allows for FTQCs with the number of error events in the order of unity $N_e\sim 1$, which is shown as a solid black line, to execute applications that originally require a much smaller error-event count. For example, at the beginning and the end of the early FTQC regime, our technique allows simulating applications (white and yellow circles with black rims) with the relaxed hardware requirement (white and yellow circles with red rims).

Figure 2

Schematic figure of the Pauli frame. The recovery operations are not physically applied to quantum computers but rather are stored in the Pauli frame and efficiently updated after each Clifford gate operation. The measurement outcomes are flipped depending on the state of the Pauli frame.

Figure 3

Schematic figure for the Pauli frame incorporating QEM. If a QEM recovery operation is a Pauli operation, it is not directly applied to the quantum computer but rather the Pauli frame is updated instead. The parity is also updated in accordance with the generated recovery operations of QEM. Here, we denote the parity corresponding to the QEM recovery operation as $p_a$ in the figure. If a QEM recovery operation is not a Pauli operator, it is performed physically. Then measurement outcomes are then postprocessed depending on the Pauli frame, parity, and QEM cost.

Figure 4

Logical error probability and QEM cost for $d$-cycle syndrome measurements plotted against the physical error rates at several code distances. (a) Logical error probability as a function of physical error rate. (b) The same figure enlarged around the threshold value. (c) QEM costs for $d$-cycle idling operations. The first-order approximations from Eq. (13) are shown as solid lines.

Figure 5

(a) Histogram of expectation values for 100-qubit random Clifford circuits. (b) Histogram of expectation values with QEM for 100-qubit random Clifford circuits. (c) Sample averages and standard deviation of 100-qubit random Clifford circuits are plotted as a function of code distances.

Figure 6

Approximation error and QEM cost of improved Solovay-Kitaev method calculated for Haar-random unitary operations. Each color corresponds to the maximum count of $T$ gates. (a) Histogram of approximation errors due to the Solovay-Kitaev decomposition. (b) Histogram of QEM costs. (c) Histogram of QEM costs for approximation errors as a function of the number of allowed $T$ gates.

Figure 7

Schematic figure of simulated seven-qubit swap test circuit. We calculate the overlap of randomly generated states and the approximation of the generated state by using the Solovay-Kitaev algorithm. The random circuit is composed of three layers, each of which consists of random single-qubit rotation gates and cnot gates acts on two randomly chosen qubits.

Figure 8

Histogram of expectation values of swap test with approximation errors. (a) Histogram of expectation values. (b) The same figure enlarged around the ideal expectation value. (c) Sample averages and standard deviation as a function of the allowed number of $T$ gates.

Figure 9

Histogram of expectation values with estimation errors. (a) Histogram of expectation values. (b) Sample averages and standard deviation as a function of estimation accuracy.

Figure 10

(a) The relation between the required number of gates for executing algorithm $N_G$ and the required code distance for encoding the information of qubit $d$. The relation is plotted for several numbers of allowed logical error counts during FTQC $N_e$. (b) The ratio between the number of required physical qubits per logical qubit with and without QEM.

Figure 11

Schematic diagram of nine logical qubits for $d=5$ surface code. The vertices of the blue and red squares correspond to physical qubits. Each boldly colored patch consisting of blue and red squares represents a single logical qubit. In each logical qubit, the bold red (blue) squares represent a Pauli-$Z$ (Pauli-$X$) stabilizer operator acting on their vertices.

Figure 12

Diagonal elements of Pauli transfer matrix for a noise map of a single logical qubit during $c$ syndrome-measurement cycles plotted versus the number of code cycles. Each color corresponds to the code distance of a logical qubit. The circles are numerical results, and the dashed lines are fitting results with an exponentially decaying function.

Figure 13

The diagram of circuit conversion from the original decision problem to the fully quantum picture. Here, $\rho$ is an $n$-qubit initial state, $\mathcal{E}$ is a noisy process, $\mathcal{D}$ is a perfect dephasing noise map, $f$ is a classical binary function on an $n$-bit string $x$, and $U_f$ is a quantum circuit that simulates $f$ as $U|0⟩|x⟩=|f(x)⟩|x⟩$. See the main text for details.