Quantum Error Mitigation and Its Progress

1.1 Extrapolation methods

As the name suggests, extrapolation methods estimate the ideal error-free calculation result by extrapolating multiple measurement results [3, 5]. They are simple yet powerful methods used in many experiments. An overview of extrapolation methods is shown in Fig. 2. The horizontal axis shows the error rate and the vertical axis shows the result (the expectation value of an observable). Of course, we cannot freely reduce the error rate, but it is relatively easy to increase calculation errors. For example, it is possible to increase the frequency of errors by slowly carrying out gate operations or by carrying out extra gate operations. By extrapolating the original calculation result and calculation results associated with increased error rates, the ideal error-free calculation result is then estimated. When extrapolation methods were first proposed, they used Richardson extrapolation with linear and polynomial functions [5]. Observing that calculation results generally decay exponentially with the frequency of calculation errors, I proposed extrapolation using an exponential function [3]. The exponential extrapolation showed extremely good performance in an actual experiment [2]. However, extrapolation methods cannot guarantee computation accuracy and can be said to be relatively heuristic.

Fig. 2. Overview of extrapolation methods.

The number of measurements, a cost factor in QEM, can be easily understood with extrapolation methods. Considering linear extrapolation as an example, for calculation error rate ε₀, we express the experimentally obtained average value of the observable as ⟨O(ε₀)⟩ and that with twice the error rate as ⟨O(2ε₀)⟩. From extrapolation, the error-mitigated result can be written as O_est = 2⟨O(ε₀)⟩ − ⟨O(2ε₀)⟩. When calculating variance, if no correlation is assumed between ⟨O(ε₀)⟩ and ⟨O(2ε₀)⟩, we obtain Var[O_est] = 4Var[⟨O(ε₀)⟩] + Var[⟨O(2ε₀)⟩]. This shows that the variance is amplified after applying QEM and that more measurements are needed to obtain the correct calculation result.

1.2 Quasi-probability method

The quasi-probability method counteracts the effect of gate noise by effectively constructing the inverse of the noise based on the noise model obtained through noise characterization techniques, such as process tomography or gate set tomography [3, 5]. We denote the quantum process corresponding to the noise as Ԑ (it may be easier to think of it as the quantum mechanical version of a transition matrix) and its inverse map as Ԑ⁻¹. While Ԑ⁻¹ can be mathematically constructed, it is not generally a “physical process” and cannot be directly operated on a quantum computer. By constructing Ԑ⁻¹ by a set of operations {B_k}_k for QEM that we can execute with fewer calculation errors, we can decompose Ԑ⁻¹ as Ԑ⁻¹ = ∑_k q_k B_k. Usually, we assume that {B_k}_k are single qubit operations.

For example, when we consider the depolarizing noise of error probability p to be , its inverse map is . Here, q₀ = (1 + , , B₀(ρ) = ρ, B₁(ρ) = XρX, B₂(ρ) = YρY, B₃(ρ) = ZρZ. Because ∑_k q_k = 1, and Ԑ⁻¹ is generally not a physical process resulting in q_k being a quasi-probability that can be negative, this method is called the quasi-probability method. A negative probability cannot be directly implemented, but the expected value “same as the case of sampling using negative probability” can be effectively calculated by post-processing of the measurement results. Consider a simple 1-qubit system as an example, which is conceptually shown in Fig. 3(a). The ideal quantum state is ρ_ideal = U |0⟩⟨0|U^†. However, because of depolarizing noise Ԑ_D, the actual quantum state is ρ_noisy = Ԑ_D (ρ_ideal). Expressing the observable to be measured as O, because the noiseless expectation value is ⟨O_ideal⟩ = q₀Tr[ρ_noisyO] + q₁Tr[Xρ_noisyXO] + q₂Tr[Yρ_noisyYO] + q₃Tr[Zρ_noisyZO], the expectation value of the observable can be measured by adding together the measurement outcomes of quantum states ρ_noisy, Xρ_noisyX, Yρ_noisyY and Zρ_noisyZ with the appropriate weight of quasi-probability. Even if a quasi-probability with a negative value exists, a non-physical inverse map can be constructed by multiplying a negative sign to measurement outcomes and performing post-processing.

It is important to implement the inverse map in a quantum circuit with multiple qubits for practical purposes. Consider when a quasi-probability method is applied to the noise Ԑ_l (l = 1,2,…N_G, where N_G is the number of gates) of multiple quantum gates. The conceptual diagram is shown in Fig. 3(b). We construct an inverse map for each error: , , , where γ^(l) is the cost coefficient. After each quantum gate (or before, depending on the formulation), operation B_k is generated with probability , and the products of sign and cost coefficient are multiplied to the measurement result. By repeating this, the average of the result gives the error-mitigated result. Because the variance of the calculated result is approximately amplified by compared with the case without error mitigation, an exponentially large number of measurements according to the number of gates is required.

Fig. 3. Overview of the quasi-probability method.

Although a suitable error-characterization method has not been proposed when the quasi-probability was first proposed, I found that gate-set tomography is an efficient error-characterization method for this method [3]. I also discovered a set of operations {B_k}_k for QEM that allows for removal of arbitrary computational errors [3]. I have also shown that the quasi-probability method can be applied not only to gate models but also temporally continuous noise models such as those described by the Lindblad master equation , extending QEM to analog quantum systems [10].

1.3 Virtual distillation method

Virtual distillation executes QEM by preparing multiple copies of a noisy quantum state ρ_noisy, executing entanglement measurements between them and post-processing the results using a classical computer. This enables us to simulate the error-suppressed quantum state as if we distill a noiseless quantum [6]. An example of the “classical” counterpart of this method is as follows: we ask several students to solve the same problem, e.g., elementary school students who often get erroneous results for simple arithmetic problems. Only when all the calculation results are the same is the answer submitted; otherwise, the results are discarded (Fig. 4). The more students involved in calculating the result, the higher the percentage of the correct answer. However, the probability of success (i.e., the probability of all students calculating the right answer) decreases exponentially with the number of students.

Fig. 4. Overview of virtual distillation method.

With virtual distillation, we can calculate the expected value of the physical quantity corresponding to the distilled quantum state , where n is the number of copies of the noisy quantum state. When we have the spectral decomposition ρ_noisy = ∑_k p_k |ψ_k⟩⟨ψ_k| (p₀ ≥ p₁ ≥ …), it is expected that the eigenstate corresponding to the largest eigenvalue is a good approximation of the ideal quantum state when the noise is small. Now, as n increases, ρ_vd asymptotically approaches |ψ₀⟩. The contribution of |ψ_k⟩ (k = 1, 2, …) is suppressed exponentially with respect to n. However, the number of required measurements increases exponentially with n. The advantage of this method is that it can mitigate errors with high accuracy if the errors are stochastic, even without information about the error model. However, coherent errors caused by rotation errors of quantum gates and the insufficient expression capability due to the lack of depth of ansatz quantum circuits in variational quantum eigensolver cannot be mitigated with this method, no matter the increase in the number of copies.

1.4 Subspace expansion method

The subspace expansion method constructs a projector (strictly speaking, this operator does not satisfy the mathematical properties of a projector but is called one here for convenience) [7]. Consider a case in which the actual quantum state immediately before measurement differs from the ideal quantum state because of noise. For example, the variational quantum eigenvalue solver is a method for determining the ground state ρ_G = |G⟩⟨G| of molecules, etc; however, the actual quantum state may be another quantum state ρ_noisy because of errors. If we can construct the projector onto the ground state P_G = |G⟩⟨G|, an error-free quantum state can be obtained as = ρ_G with p_G being the projection probability (Fig. 5(a)). However, because |G⟩ is an extremely large quantum state in reality, the expression of P_G cannot be obtained in the first place, and the projection cannot be executed accurately. We thus seek to construct a projection operator (which strictly speaking, does not satisfy the mathematical properties of a projection operator but called one here for convenience) that can project the noisy state onto a space with the lowest possible energy. Using Pauli operators P_k to express such a projection operator as (where c_k is a complex number), we optimize {c_k}_k using a classical computer so that the energy of the projected quantum state (where p is the projection probability) can be minimized (Fig. 5(b)). What P_k to choose is arbitrary. Methods for constructing P_k from, for example, excitation operators of a molecule’s spin-orbitals, have been proposed [7]. This method can suppress coherent errors to a certain extent but is known to be unsuitable for suppressing stochastic errors such as bit flips.

Fig. 5. Overview of subspace expansion methods.

2.1 Application of QEM to quantum sensing

NTT has developed the world’s first framework for quantum sensing incorporating QEM. Quantum sensing is an active area of research in the field of quantum information that uses quantum states to efficiently probe fields such as magnetic fields one wishes to measure. This is done by interacting the quantum states with the field followed by the readout. The process is repeated and the results are accumulated to estimate the value of the magnetic field. What is important about quantum sensing is that when quantum entangled states are used as probes, quantum advantageous scaling can be achieved depending on the number of qubits N. However, if the noise fluctuates at each time of measurement, systematic errors occur in the accumulated value and estimated magnetic-field value, and quantum advantages cannot be achieved (Fig. 6(a)). Our research group has shown that even when noise fluctuates each time the quantum device is executed, virtual distillation can act as a “filter” that removes such noise and accurately mitigates systematic errors [8]. We have also shown that quantum advantageous scaling can be restored (Fig. 6(b)).

Fig. 6. Effect of QEM (virtual distillation) on quantum sensing.

2.2 Generalized subspace expansion method

Our research group proposed the generalized subspace expansion method, which is quite a general QEM method, which includes subspace expansion and virtual distillation as special cases [9]. I stated above that in subspace expansion, the projection operator can be optimized so that energy is minimized. The essence of the generalized subspace expansion method is extending P_k to extremely general operators. More specifically, quantum states (and more complex operators that include them) are used as P_k. For example, taking P₀ = I, P₁ = ρ_noisy, the projected state is , and the expected value of the observables corresponding to an error-mitigated quantum state expanded by a series of powers of a noisy quantum state can be obtained. We call this method the power subspace method (Fig. 5(c)). Our research group also proposed the fault-subspace method that uses the essence of extrapolation methods [5] in the construction of projectors. Unification of the power subspace method and fault-subspace method is also possible. The generalized subspace expansion method inherits the advantages of both subspace expansion and virtual distillation methods and can mitigate both coherent errors and stochastic errors with high accuracy. Therefore, far more accurate QEM is made possible compared with subspace expansion or virtual distillation alone.