Transmitter and method for determining perturbation vector
Patent Information
- Application Number
- PCT/SE2025/051170
- Authority / Receiving Office
- WO · WO
- Patent Type
- Applications
- Current Assignee / Owner
- Priority Date
- 2025-03-24
- Filing Date
- 2025-12-19
- Publication Date
- 2026-10-01
Smart Images
Figure SE2025051170_01102026_PF_FP_ABST
Abstract
Description
[0001] TRANSMITTER AND METHOD FOR DETERMINING PERTURBATION VECTOR
[0002] TECHNICAL FIELD
[0003] The disclosure relates to a Multiple-Input Multiple-Output (MIMO) transmitter and a method performed by the MIMO transmitter. A corresponding computer program and a computer readable medium are also disclosed.
[0004] BACKGROUND
[0005] Multiple-Input Multiple-Output (MIMO) downlink data transmission improves wireless communication by using multiple antennas to transmit and receive data simultaneously, increasing speed and efficiency without requiring extra bandwidth. MIMO enhances signal reliability by reducing interference and fading, leading to more stable connections and better coverage. In a multi-user MIMO downlink data transmission, precoding techniques can be used to reduce inter-user interference and allow users to detect their respective data non-cooperatively, minimizing error-rate and maximizing throughput.
[0006] Vector Perturbation Precoding (VPP) is a widely studied non-linear precoding technique that performs transmit-side channel inversion over a perturbed user data vector to reduce the transmit power scaling. Although VPP has been shown to achieve better error performance compared to other precoding techniques, finding an optimal perturbation vector for user data in VPP is known to be NP-hard, making its implementation in massive / large MIMO systems to be infeasible. Quantum computing metrics may be used for finding an optimal perturbation vector, as they offer a computational advantage by using principles of quantum mechanics for solving an optimization problem such as finding the optimal perturbation vector.
[0007] Optimization problems have been solved for quantum computers by transforming the optimization problem into an Ising Hamiltonian or Quadratic Unconstrained Binary Optimization (QUBO) formulation and using Quantum Approximate Optimization Algorithms (QAOA) to determine a ground state of an Ising Hamiltonian corresponding to a configuration of decision variables. However, noisy intermediate¬ scale quantum computers encounter challenges due to noise, resulting in lowaccuracy in solutions and prolonged runtime when tackling problems with high circuit depth and substantial number of variables. Further, variational quantum algorithms such as the QAOA to solve optimization problems such as combinatorial optimization problems on near-term quantum hardware face limitations in circuit depth and number of variables.
[0008] M. H. Munoz-Arias, S. Kourtis, and A. Blais, “Low-depth Clifford circuits approximately solve MaxCut”, Physical Review Research 6(2), 023294, APS, 2024, relates to a quantum-inspired approximation algorithm known as ADAPT-Clifford algorithm for MaxCut, wherein the quantum-inspired approximation algorithm is based on low-depth Clifford circuits, e.g., in case of. dense optimization problems. The quantum-inspired approximation algorithm can be applied to zero-field Ising models (ZFIMs), consisting solely of two-body interactions with no longitudinal-field terms. However, the general QUBO model maps into the longitudinal-field Ising models (LFIMs) which also includes single-spin terms. The ADAPT-Clifford algorithm described in “Low-depth Clifford circuits approximately solve MaxCut” is not designed to include the single-spin terms and solve the LFIMs. A reason is that single-spin term gradients evaluated in the ADAPT-Clifford algorithm vanish identically. In a vector perturbation precoding (VPP) scenario, this may lead to a suboptimal selection of a precoding vector, wherein the suboptimal selection may lead to an ineffective interference cancellation at a MIMO receiver and may require a high transmission power when transmitting symbols in a MIMO data transmission for maintaining a signal quality.
[0009] Kasi, Srikar, et al., " Quantum annealing for large MIMO downlink vector perturbation precoding", ICC 2021 -IEEE International Conference on Communications, IEEE, 2021, relates to a processing architecture for a vector perturbation precoding (VPP) problem based on Quantum Annealing (QA).
[0010] SUMMARY
[0011] An object of the invention is to improve determining of a perturbation vector in a Multiple-Input Multiple-Output (MIMO) system comprising a MIMO transmitter and a MIMO receiver.
[0012] A first aspect of the invention relates to a method performed by a MIMO transmitter. The method comprises acquiring at least one user data symbol to be transmitted to aMIMO receiver. The method further comprises acquiring a precoder matrix representing characteristics of radio channels between the MIMO transmitter and the MIMO receiver. The method further comprises determining a Quantum Unconstrained Binary Optimization (QUBO) representation for a transmission power required for transmitting the user data symbol. The QUBO representation is determined based on the at least one user data symbol and the precoder matrix. The method further comprises embedding the QUBO representation on a quantum precoder comprising a plurality of qubits and at least one ancilla qubit. The method further comprises processing the embedded QUBO representation using an ADAPT-Clifford algorithm until a configuration of the plurality of qubits and the at least one ancilla qubit is reached which minimizes the transmission power required for transmitting the user data symbol. The method further comprises determining a perturbation vector based on a final configuration of the plurality of qubits. The method further comprises transmitting the at least one user data symbol using the precoder matrix and the perturbation vector. The at least one user data symbol is transmitted to the MIMO receiver.
[0013] A second aspect of the invention relates to a MIMO transmitter. The MIMO transmitter is configured to acquire at least one user data symbol to be transmitted to a MIMO receiver. The MIMO transmitter is further configured to acquire a precoder matrix representing characteristics of radio channels between the MIMO transmitter and the MIMO receiver, determine a QUBO representation for a transmission power required for transmitting the user data symbol. The QUBO representation is determined based on the at least one user data symbol and the precoder matrix. The MIMO transmitter is further configured to embed the QUBO representation on a quantum precoder comprising a plurality of qubits and at least one ancilla qubit. The MIMO transmitter is further configured to process the embedded QUBO representation using an ADAPT-Clifford algorithm until a configuration of the plurality of qubits and the at least one ancilla qubit is reached which minimizes the transmission power required for transmitting the user data symbol. The MIMO transmitter is further configured to determine a perturbation vector based on a final configuration of the plurality of qubits. The MIMO transmitter is further configured to transmit the at least one user data symbol using the precoder matrix and theperturbation vector. The at least one user data symbol is transmitted to the MIMO receiver.
[0014] A third aspect of the invention relates to a computer program. The computer program comprises instructions which, when executed on processing circuitry of a MIMO transmitter, cause the processing circuitry of the MIMO transmitter to carry out the method according to the first aspect.
[0015] A fourth aspect of the invention relates to a computer-readable medium. The computer-readable medium comprises instructions that, when executed by processing circuitry of a MIMO transmitter, cause the processing circuitry of the MIMO transmitter to carry out the method according to the first aspect.
[0016] BRIEF DESCRIPTION OF THE DRAWINGS
[0017] Figure 1 is a schematic diagram illustrating an example of an environment in which embodiments presented herein can be applied.
[0018] Figure 2 is a flowchart illustrating a method performed by the MIMO transmitter.
[0019] Figure 3 is an example setup of a MIMO transmitter and a MIMO receiver. Figure 4 shows an example of a communication system.
[0020] Figure 5 is a graph illustrating a ratio of an transmission power difference as a function of the number of antennas of the MIMO transmitter.
[0021] Figure 6 illustrates two graphs illustrating a bit error rate (BER) as a function of a signal-to-noise ratio.
[0022] Figure 7 illustrates a block diagram illustrating embodiments of the MIMO transmitter.
[0023] DETAILED DESCRIPTION
[0024] Figure 1 schematically illustrates an example environment in which embodiments presented herein can be applied. Figure 1 illustrates a MIMO transmitter 100 and a MIMO receiver 120, which together form a MIMO system 150. The MIMOtransmitter 100 comprises a quantum precoder 130. The MIMO transmitter 100 and the MIMO receiver 120 may be communicatively connected via a communication path 140 representing radio channels between the MIMO transmitter 100 and the MIMO receiver 120 using wireless communications, e.g., Wi-Fi, and / or a cellular network corresponding to one of or a combination of 5G cellular networks, Long Term Evolution (LTE) and Evolved Packet System (EPS), LTE-advanced, Universal Mobile Telecommunications System (UMTS), or any other current or future wireless network, such as a future 3GPP 6G network. Alternatively, the quantum precoder 130 may be a separate entity provided separately to the MIMO transmitter 100. The quantum precoder 130 comprises a plurality of qubits (short for quantum bits) and at least one ancilla qubit.
[0025] Figure 2 is a flowchart illustrating a method 200 performed by the MIMO transmitter 100, as shown in Figure 1 and described in the text thereto. The method 200 comprises acquiring 210 at least one user data symbol to be transmitted to the MIMO receiver 120. The method 200 further comprises acquiring 220 a precoder matrix. The precoder matrix represents characteristics of radio channels in the communication path 140 between the MIMO transmitter 100 and the MIMO receiver 120. The method 200 further comprises determining 230 a QUBO representation for a transmission power required for transmitting the user data symbol. The determining 230 is based on the at least one user data symbol and the precoder matrix. The method 200 further comprises embedding 240 the QUBO representation on a quantum precoder 130. The method 200 further comprises processing 250 the embedded QUBO representation using an ADAPT-Clifford algorithm until a configuration of the plurality of qubits and the at least one ancilla qubit is reached which minimizes the transmission power required for transmitting the user data symbol. The method 200 further comprises determining 260 a perturbation vector based on a final configuration of the plurality of qubits. The method 200 further comprises transmitting 270 the at least one user data symbol using the precoder matrix and the perturbation vector to the MIMO receiver 120.
[0026] The solution presented herein addresses the above-mentioned challenges of enabling an improved determination of a perturbation vector. Embodiments of the invention are based on the understanding that the MIMO transmitter 100 may solvethe optimization problem of the determination of the perturbation vector by adding the at least one ancilla qubit, thereby making the QUBO representation processable using the ADAPT-Clifford algorithm. Advantageously, the MIMO transmitter 100 may determine an optimal perturbation vector with improved accuracy requiring a reduced transmission power and / or resulting in an increased signal quality. In comparison to usage of a QA, the solution presented herein requires lower computational resources. By acquiring 210, 220 the at least one user data symbol and the precoder matrix, the MIMO transmitter 100 may process the user data symbol using the precoder matrix optimizing a transmission across multiple antennas. By determining 230 and embedding 240 the QUBO representation, the MIMO transmitter 100 may process the user data symbol and the precoder matrix, e.g., using a subset of quantum phenomena or using classically stimulable quantum phenomena. By processing the embedded QUBO representation, the perturbation vector may be determined based on the final configuration of the plurality of qubits.
[0027] Referring to Figure 2, in determining 230 the QUBO representation, the QUBO representation may be expressed with:
[0028]
[0029] i i*k
[0030] wherein Pt(x) corresponds to the transmission power, x corresponds to a vector of binary decision variables, and Q corresponds to a QUBO matrix defining how the binary decision variables interact with each other. The transmission power corresponds to a cost to be minimized.
[0031] Embedding 240 the QUBO representation may comprise sending, to the quantum precoder 130, the QUBO representation, the LFIM, or the enlarged ZFIM, e.g., if the quantum precoder 130 is provided separately to the MIMO transmitter 100. In this example, the quantum precoder 130 receives the QUBO representation, the LFIM, and / or the enlarged ZFIM.
[0032] The embedding 240 may comprise determining 242 a LFIM representation based on the QUBO representation. Optionally, the MIMO transmitter 100 maps the QUBOrepresentation to the LFIM, by mapping the vector of binary decision variables x e {0,1} to a set (herein referred to as s e {±1}), such as a spin configuration, using
[0033] The LFIM may be expressed as:
[0034] ?t ~ ’ ^ik Si$k " I” ’ hisi> Eq. 2
[0035]
[0036] i*k i
[0037] wherein
[0038]
[0039] h[ = Qu / 2 and = Qtk / 4. The first term of the LFIM,
[0040] i*kwiksisk’maY represent an interaction strength between a qubit i and a qubit k of the plurality of qubits in the quantum precoder 130 (alternatively, a spin i and spin k), wherein wikmay represent how qubit i and qubit k interact with each other. In an example, the interaction between qubit i and qubit k, represent crosschannel interference or cannel coupling in the radio channels between the MIMO transmitter and MIMO receiver. The second term of the LFIM, i hisi, may represent a longitudinal field strength applied to qubit i representing an external field applied along a longitudinal axis.
[0041]
[0042] and skcorrespond to spin configurations of the qubit i and the qubit k, respectively. The number of the plurality of qubits (herein referred to as T) depends on a modulation order used for transmitting the at least one data symbol. For example, a qubit may represent two symbols in a signal constellation of a modulation scheme with a modulation order. In an example, two qubits are required to represent 4 possible symbols in a signal constellation of a Quadrature Phase Shift Keying (QPSK) modulation scheme. In another example, one qubit is required to represent 2 symbols in a signal constellation of a Binary Phase Shift Keying (BPSK) modulation scheme. In another example, 4 qubits are needed to represent 16 symbols in a signal constellation of a 16-Quadrature Amplitude Modulation (QAM) modulation.
[0043] Embedding 240 the QUBO representation may comprise embedding 244 the LFIM representation in an enlarged Zero Field Ising Model (ZFIM). Embedding 244 the LFIM representation in an enlarged ZFIM may be performed by introducing the atleast one ancilla qubit (ancilla or auxiliary spin) in the quantum precoder 130. The at least one ancilla qubit may represent the longitudinal field strength applied to the qubit i. The at least one ancilla qubit may be coupled with the second term of LFIM model. In the enlarged ZFIM, the external field strength of the ancilla qubit is reused as a coupling strength. The at least one ancilla qubit may be represented with the second term of the LFIM. For example, if the quantum precoder comprises only one ancilla qubit, the enlarged ZFIM may be expressed as:
[0044] Eq. 3
[0045]
[0046] wherein W[ ■r+1= h[. The number of the plurality of qubits and the ancilla qubit may thus correspond to T + 1. Embedding the enlarged ZFIM may correspond to a stabilizer subspace embedding.
[0047] Alternatively, or additionally, embedding 240 the QUBO representation may comprise preparing 246, via the quantum encoder, the plurality of qubits and the at least one ancilla qubit.
[0048] Optionally, processing 250 the embedded QUBO representation comprises applying the Adapt-Clifford algorithm on the enlarged ZFIM representation.
[0049] The Adapt-Clifford algorithm may be an algorithm for searching through a Clifford manifold by combining a minimal set of generating elements, e.g., of a Clifford group such as a controlled-NOT gate, a Hadamard gate, and phase gate. The algorithm may prepare a stabilizer state starting from the randomly selected qubit, and grow the stabilizer state qubit by qubit, in such a way that at a certain step the state is a product of an entangled state of all the active qubits and the product state of all inactive qubits. After a number of steps which is equal to the number of the plurality of qubits and the at least one ancilla qubit, a measurement in a computational basis, i.e. in basis of a Hilbert space, may be performed.Optionally, processing 250 comprises a applying 254 via the quantum encoder a Clifford Gate to each of the plurality of qubits and the at least one ancilla qubit based on the QUBO representation.
[0050] For applying the Adapt-Clifford algorithm, an initial state (herein referred to as |<p')) may be prepared, based on the enlarged ZFIM, using the quantum precoder 130 as: \
[0051]
[0052] (p') = 10)®T+1. The quantum precoder 130 may start with the j'th qubit and grow the initial state qubit by qubit of the plurality of qubits and the ancilla qubit, i.e., the resulting state is evolved iteratively. A qubit is marked as active if the Clifford Gate has been applied to the qubit for ensuring that a qubit of the plurality of qubits and the ancilla qubit has not been used twice in the Adapt-Clifford algorithm. Step r is iteratively performed until each a plurality of steps T + 1 a qubit is marked as active. A qubit to which no gate has been applied yet is marked as inactive. The active qubits are represented by indices of a vector a^rwhile inactive qubits are represented by indices of a vector b^r. The vectors
[0053]
[0054] and b^ store positions of all active and inactive qubits at the step r. At the step r the initial state evolved to an entangled state of all active and inactive qubits in a product state. The initial state at step r = 0, the product states, and the resulting stabilizer state at step r = T, may correspond to stabilizer states. Using stabilizer states during applying the Adapt-Clifford algorithm may enable an efficient computation. A use of stabilizer states when simulating stabilizer circuits may be efficient, because the stabilizer states are described by stabilizer generators requiring only polynomial space, unlike the exponential growth of the full quantum state vector. Further, a restricted set of operations (e.g., Pauli gates, CNOT gates, and / or Hadamard gates) may modify or introduce stabilizer generators without exponential information growth, and measurements in the computational basis may yield deterministic outcomes, as stabilizer states are eigenstates of the stabilizer generators.
[0055] At the step r = 0, the qubit k may be randomly selected and the product state at r = 0 ( herein referred to as |<0)is prepared as |<p0) = ZkHT+1\0)T+1, wherein H corresponds to the Hadamard gate and Zkcorresponds to the Pauli Z gate appliedon the fcth qubit. The kth qubit is an active qubit, i.e., a0= {k} and a rest of qubits of the plurality of qubits and the ancilla qubits to inactive qubits, i.e.,
[0056]
[0057] =
[0058] {1. T + 1}\{ / C).
[0059] At a step r = 1, a maximum gradient at r = 1 may be estimated analytically. A gradient, such as the maximum gradient, at steps r > 1, may correspond to a largest coefficient of all possible two qubit interactions in the Hamiltonian, each comprising of an active and an inactive qubit. Thus, the maximum gradient may be a measure for finding a qubit-pair candidate with the greatest influence on reducing the transmission power. Based on the maximum gradient, an yth qubit of the plurality of qubits and the ancilla qubit is selected. The Clifford gate may be applied to the yth qubit. Thus, a product state (herein referred to
[0060]
[0061] as at r = 1 is: \(p^) =
[0062] . IT
[0063] e
[0064]
[0065] 1^7 kZkH®T+1\0)®T+1, wherein Z7correspond to the Pauli Z gate applied on the jth qubit and Ykcorresponds to a Pauli Y gate on the fcth qubit. The active qubits correspond to the fcth and yth qubit,
[0066]
[0067] i.e., = [k, y) and = {1,..., T + 1}\
[0068] At steps r = 2,... T a pair of qubits are selected with a maximal gradient at step r.
[0069] (r)
[0070] The gradient at step r (herein referred to as g (r- )
[0071]
[0072] may be computed with
[0073] Z Eq. 4wl,b(-r~1') )r-l>
[0074]
[0075] I
[0076] wherein Xb(r-1)corresponds to the Pauli X gate on the b(r-1)th qubit.
[0077] Based on the maximal gradient at step r, the next qubit I is selected by breaking a tie arbitrarily, and the Clifford gate is applied to qubit I.
[0078] After T + 1 steps are completed, the resulting state corresponds to:r T e$zi YkZkH®T+11 Q^OT+I Eq. 5
[0079]
[0080] Lr=2
[0081] After completion of all steps, a measurement of the resulting state may represent a bit string comprising the decision variables X[.
[0082] Determining 260 the perturbation vector may comprise acquiring 262 the final configuration of the plurality of qubits from the quantum precoder 130.
[0083] The resulting state comprises the states of the plurality of qubits and the at least one state of the at least one ancilla qubit. The final configuration corresponds to the states of the plurality of qubits only, excluding the at least one state of the at least one ancilla qubit.
[0084] If the quantum precoder 130 is comprised in the MIMO transmitter 100, acquiring 262 the final configuration of the plurality of qubits may comprise determining, from the quantum precoder 130, the final configuration of the plurality of qubits.
[0085] Alternatively, if the quantum precoder 130 is provided separately to the MIMO transmitter 100, acquiring 262 the final configuration of the plurality of qubits may comprise sending, by the quantum precoder 130 to the MIMO transmitter 100, the final configuration of the plurality of qubits or a representation thereof, and receiving, by the MIMO transmitter 100 from the quantum precoder 130, the configuration of the plurality of qubits.
[0086] Determining 260 the perturbation vector may comprise determining 264 the complex elements of the perturbation vector v based on the final configuration of the plurality of qubits.
[0087] Transmitting 270 the at least one user data symbol may comprise sending at least one modified user data symbol u’ to the MIMO receiver 120. The at least one modified user data symbol may be determined, e.g., computed, by the MIMO transmitter 100 based on the precoder matrix P as following:u' = Pu + v.
[0088] The MIMO transmitter 100 may send the at least one modified user data symbol, to the MIMO receiver 120, using transmit antennas of the MIMO transmitter 100.
[0089] In a first embodiment, acquiring 210 the user data symbol comprises receiving the user data symbol u from, e.g., a core network of a communication network or from a processor device in the MIMO transmitter, wherein the MIMO transmitter 100 is comprised in the communication network.
[0090] In the first embodiment, acquiring 220 the precoder matrix P comprises determining the precoder matrix based on Channel State Information (CSI).
[0091] In the first embodiment, the method 200 comprises determining 230 the QUBO representation.
[0092] In the first embodiment, the user data symbol u gets perturbed, wherein a perturbation during transmission is represented by a perturbation vector v. The perturbation vector v may be a vector of Gaussian integers. The perturbed transmit vector d is:
[0093] d = u + TV, Eq. 7
[0094] where τ = 2 (|cmax| + Δ / 2) may be a constant chosen to provide symmetric
[0095]
[0096] decoding regions around constellation symbols in a modulation scheme used. |cmax| corresponds to a magnitude of a largest constellation symbol and Δ to a spacing between the constellation symbols of the modulation scheme used. For example, the modulation scheme may correspond to BPSK, QPSK, QAM, 64-QAM, 256-QAM, 1024-QAM etc. When the MIMO transmitter 100 transmits the user data symbol, a symbol vector may be transmitted. The transmitted symbol vector x is
[0097] x = Pd / √P_t, Eq. 8wherein Pt= ||Pd||2corresponds to a transmission power scaling factor at the MIMO receiver. The MIMO receiver may receive a symbol vector y (herein referred to a received symbol vector). The received symbol vector may be defined as
[0098] y = 1 / √P_t ( HPd) + n, Eq. 9
[0099]
[0100] wherein n corresponds to wireless channel noise and H corresponds to the wireless channel matrix.
[0101] Thus, in the first embodiment, for minimal transmission power an optimal perturbation vector in a vector precoding problem may be formulated in the following QUBO representation as:
[0102] v' = argmin_v ||H^H (HH^H)^{-1}(u + τv)||², Eq. 10
[0103] wherein v' corresponds to the optimal perturbation vector leading to a minimal transmission power required for transmitting the user data symbol. The MIMO transmitter 100 comprises a number M of transmitter antennas, and the MIMO receiver 120 comprises a number N of receiver antennas. The vector precoding problem aims to minimize the transmission power required for transmitting the given user data symbol u through the wireless channel H E
[0104]
[0105] by perturbing it with the optimal perturbation vector v'. In an example, u and v belong to a constellation subspace SM. The constellation subspace is denoted as a signal constellation of a plurality of mapped user data symbols, wherein the user data symbol is one of the plurality of user data symbols. The constellation subspace varies based on the number of transmitter antennas M and on the choice of the modulation scheme of the signal constellation.
[0106] In the first embodiment, the optimal perturbation vector may be expressed as:v' = argminv\\A(u + TV)||2, Eq. 11
[0107] w
[0108]
[0109] ith 4 =
[0110] The term A(u + TV) may be further formulated as:
[0111] Eq. 12 A(u + τv) =
[0112]
[0113] =
[0114] + wf) - + wf
[0115] MieN + wf) + SvieJV^iiW +Tpf )
[0116]
[0117] ^i\ / ieN + XvieA^
[0118] leading to ||4(t + w)||2= (^SV / CEMI^ + w)|2}.
[0119]
[0120] In the first embodiment, embedding 240 the QUBO representation comprises embedding the QUBO representation in the enlarged ZFIM representation and embedding the enlarged ZFIM representation on the quantum precoder 130. In the first embodiment, the quantum precoder 130 is a classical computer with access to a classical solver. In the first embodiment, advantageously, no quantum annealer is needed and the ADAPT-Clifford algorithm may be performed on the classical hardware. Embedding may comprise selecting the number of qubits based on the modulation order used.
[0121] In the first embodiment, processing 250 the QUBO representation comprises processing the QUBO representation until the final configuration of the plurality of qubits and the at least one ancilla qubit is determined which minimizes the transmission power required for transmitting the user data symbol. In an example, the processing 250 may comprise processing the enlarged ZFIM representation. In the first embodiment the final configuration of the plurality of qubits correspond to a plurality of decision variables (herein referred to as xi). Further, the final configuration of the plurality of qubits and the at least one ancilla qubit corresponds to the resulting state as described in Eq. 5, i.e., stabilizer states of each qubit of the plurality of qubits and the at least one ancilla qubit. Processing the enlarged ZFIM may comprise determining the decision variables using the classical computer with access to the classical solver. The plurality of qubits and the at least one ancilla qubit and applying the Clifford Gates are simulated using a simulator. The simulator may be a fast stabilizer circuit simulator, e.g., using a stabilizer tableau representation such as Stim (Gidney, Craig, " Stim: a fast stabilizer circuit simulator" Quantum 5, 497, 2021). The simulator may efficiently simulate the resulting state and the resulting quantum circuit in a binary representation on the classical computer.
[0122] In the first embodiment, determining 260 the perturbation vector is based on the states of the plurality of qubits. The at least one state of the at least one ancilla qubit is omitted for determining the perturbation vector.
[0123] In the first embodiment, determining 260 the perturbation vector comprises acquiring 262 the states of the plurality of qubits from the classical solver.In the first embodiment, determining 260 the perturbation vector comprises determining 264 the complex elements of the perturbation vector based on the decision variables of the plurality of qubits.
[0124] In the first embodiment, ||A(u + τv)||2corresponds to a complex modulus or a magnitude of the complex vector A(u + τv) as described in Eq. 11. Therefore, the term ||A(u + τv)||2results in:
[0125] ||A(u + τv)||2
[0126]
[0127] wherein viRcorresponds to the real part of the perturbation vector v, and viIto the imaginary party of the perturbation vector v. The real and the imaginary part of the perturbation vector may be binarized based on the modulation scheme used. Z. I. Tabi, et.al., " Evaluation of Quantum Annealer Performance via the Massive MIMO Problem", in IEEE Access, vol. 9, pp. 131658-131671, 2021, doi:
[0128] 10.1109 / ACCESS.2021.3114543, refers to a modulation scheme bit mapping as shown in Table 1, wherein the modulation scheme bit mapping is based on applying a Maximum Likelihood (ML) detection problem to the modulation schemes.
[0129] BPSK vi→ (2xi- 1)
[0130] QPSK vi→ (2x2i-1- 1) + j(2x2i- 1)
[0131] 16-QAM vi→ (4x4i-3+ 2x4i-2- 3) + j(4x4i-1+ 2x4i- 3)
[0132]
[0133] 64-QAM vi→ (8x6i-5+ 4x6i-4+ 2x6i-3- 7) + j(8x6i-2+ 4x6i-1+ 2x6i- 7)
[0134]
[0135] Table 1: Mapping between modulation scheme and perturbation vector In the first embodiment and in correspondence with Table 1, the perturbation vector may be determined using the mapping as presented in Table 1.
[0136] Alternatively, determining the perturbation vector by using the mapping as presented in Table 1, the real and imaginary part of the perturbation vector vi= viR+ jviI, may be approximated as:
[0137]
[0138] viR, viI= ci∑k=0r-12-kxk,i- di∈ [-di, 2ci- di), Eq. 14
[0139] wherein j corresponds to the fc’th binary decision variable of r binary variables, determined during the ADAPT-Clifford algorithm in T steps. Further, ci, dimay be modelled vectors, wherein the vectors are modelled based on a user-set range for values of the imaginary and real port of the perturbation vector viR / viI. The userset range corresponds to the number of the plurality of qubits T. The user-set range may be set based on prior knowledge based on characteristics of the MIMO transmitter and MIMO receiver or based on a random selection of a large range with more qubits spanning the range. In an example, the vectors are modelled assuming a domain ci, diof each viR, viIvariable and a resolution using r qubits are defined. For example, there are 2A8 = 256 possible states, wherein each possible state D
[0140] corresponds to a discrete point in the range [-3, 3), in case viRhas a range of [-3,3) = [-3, 2*3 - 3) implying ci= 2 and di= 3, and using 8 qubits. An intermediate state |0000011⟩ in the ADAPT-Clifford algorithm (in case of 6 plurality of qubits and one ancilla qubit, and the first two qubits being active) may be in this example computed as 2(20+ 2-1) - (3) = 2(1 + 0.5) - 3 = 0.
[0141] In the first embodiment, transmitting 270 the at least one modified user data symbol determined is based on the precoder matrix and the perturbation vector.In a second embodiment, the acquiring 210 of the at least one user data symbol, the acquiring 220 of the precoder matrix and the determining 230 of the QUBO representation is performed as described for the first embodiment.
[0142] In the second embodiment, embedding 240 the QUBO representation comprises preparing 246 the plurality of qubits and at least one ancilla qubit, wherein the quantum precoder 130 is instructed by the MIMO transmitter to prepare the plurality of qubits and at least one ancilla qubit based on the QUBO representation.
[0143] In the second embodiment, the plurality of qubits may be prepared by mapping the vector component to qubits using, for example, the equations of Table 1 to encode the at least one user data symbol to the plurality of qubits. Encoding the at least one user data symbol to the plurality of qubits may utilize quantum superposition and quantum phase rotations. Additionally, at least one ancilla qubit is added and initialized.
[0144] In the second embodiment, the processing 250 of the embedded QUBO representation is performed using the ADAPT-Clifford algorithm until the configuration of the plurality of qubits and the at least one ancilla qubit is reached which minimizes the transmission power required for transmitting the user data symbol. The output of the ADAPT-Clifford algorithm is the resulting state of the plurality of qubits and the at least one ancilla qubit corresponding to the final configuration of the plurality of qubits.
[0145] In the second embodiment, Quantum Unitary gate comprising Clifford gates may be at applied to the Zth qubit at step r > 1 as:
[0146] ei4Vl=s, HmCN0TimR‘li( - - ) CNOT,mS Hm,E<15
[0147] wherein Slis a phase gate and Hmis a Hadamard gate acting on the lth qubit, CNOTl,mis a controlled-NOT gate with qubit l and qubit m as control and target qubits, respectively. Furthermore, Rx(l)(-π / 2) is based on a variant of theHadamard gate (herein referred to as HlYZ) which swaps a y axis and a z axis in a block sphere representation of lth qubit as:
[0148] Rx(l)(-π / 2) = HlYZZl. Eq. 16
[0149]
[0150] xk 2'1
[0151] In the second embodiment, the determining 260 of the perturbation vector is performed based on the states of the plurality of qubits comprised in the resulting state. The at least one state of the at least one ancilla qubit comprised in the resulting state is omitted for determining the perturbation vector.
[0152] In the second embodiment, determining 260 the perturbation vector comprises acquiring 262 the states of the plurality of qubits from the quantum precoder 130. Alternatively, the MIMO transmitter 100 acquires the decision variables from the quantum precoder 130.
[0153] In the second embodiment, the determining 264 of the complex elements and the transmitting 270 of the at least one user data symbol are performed as described for the first embodiment.
[0154] In a third embodiment, the acquiring 210 of the at least one user data symbol, the acquiring of the precoder matrix, the determining 230 of the QUBO representation, the embedding 240 of the QUBO representation, the processing of the embedded QUBO representation, the determining 260 of the perturbation vector, and the transmitting 270 of the at least one user data symbol, are performed according to the first or second embodiment, wherein the modulation scheme may be a QPSK scheme.
[0155] In the third embodiment, the QUBO representation may be formulated based on Eq.
[0156] 11, Eq. 13, and Table 1, as:||4(u + ) ||2
[0157] - A1^ (u{ + r(2x2i- 1 VieN + A1^ (uf + x(2x2i- 1)) VieN i-u? + 2A%iTX2i-1
[0158] V 2 li™2i - 4^T) Eq. 17 WieN kiui + 24fiT%2j_1kirx2i WieN H D+ 24^TX2i-! “A^iT“A'kiUli“2AkiTX2i VkeM y wieN V 2 + ^kiX)
[0159] ^kiui + 24fiTX2j_1- A^T + A^u? + 24(fTX2j
[0160]
[0161] + (A^U? - AliUf - v(Aki- 4^) )
[0162] 2>lfciT%2j_1+ 24[fTX2i
[0163]
[0164] For simplification the following is defined:
[0165] Yki= A^uf - Alkiu{ - r^Aki- A^t),
[0166] Eq. 18 Z
[0167]
[0168] kt = + A^uf - r(Aki+.
[0169] Thus, from Eq. 17 and Eq. 18 follows:
[0170]
[0171] The square of a sum is equal to the sum of the squares of all the summands plus the sum of all the double products of the summands in twos. Therefore, the determining 260 of the perturbation vector may comprise determining the perturbation vector with:Vi =i-1 2AkiTX2t + Yfct) V LkeM ■VieN N-l N + 2 ^ (2A^TX2i-1- 2^X21 i=l m=i+l + Yki^(2AkmTX2m- 1 2i4^jnTX2m+ Ffcm) \1. D ~.i „ \2 Eq. 20 WieN N-l N + 2 ^ (2A^TX2i-1+ 2^X21 i=l m=i+l
[0172]
[0173] wherein xfcorresponds to the decision variables determined from the resulting state.
[0174] In a fourth embodiment, the acquiring 210 of the at least one user data symbol, the acquiring 220 of the precoder matrix, the determining 230 of the QUBO representation, the embedding 240 of the QUBO representation, and the processing of the embedded QUBO representation, are performed as described for any of the first to third embodiments.
[0175] In the fourth embodiment, determining 260 the perturbation vector comprises determining 264 the complex elements of the perturbation vector based on the final configuration of the plurality of qubits by the quantum precoder 130. In response to determining by the quantum precoder 130 the complex elements the MIMO transmitter may acquire the complex elements of the perturbation vector from the quantum precoder 130.
[0176] In a fifth embodiment, the method 200 may be performed according to any of the first to the fourth embodiments, wherein a 2x2 MIMO system is used using QPSK modulation, wherein 2 vector components are represented by 2 qubits each. Thus, the number of the plurality of qubits is chosen to be T = 4 qubits in the fifth embodiment. By adding an ancilla qubit the initial state in the Adapt-Clifford algorithm is |<p'> = |0)®5= |+ + + + +), wherein |+) = H|0>. The states |+) and |-) =ZH\O) are eigenstates of the Pauli-x operator corresponding to eigenvalues +1 and -1, respectively. At the step r = 0, the state of the kth qubit at k = 2 is flipped such that |<p0) = |d - I- + +). At the step r = 0, the k = 2nd qubit is marked as active, e.g., in an array: [2], and the rest as inactive [1, 3,4,5]. At this point, a first set of (1) (1) (1)
[0177] gradients may be obtained with Eq. 4 as: g1,2(1)= g2,3(1)= g2,4(1)= 1. All of the gradients correspond to the maximum gradients, wherein arbitrarily the pair of qubits (2,4) is selected. The state at r = 1 may thus be obtained with:
[0178] I«P1> =e‘4Z4r2 |Vo)
[0179] Eq. 21 = 4[|+ - + + +) -!+ + + - +>]■
[0180]
[0181] V
[0182] In the fifth embodiment, the mapping of the active and inactive qubits may thus be updated to [2,4] and [1,3,5], respectively. A second set of gradients may be obtained using Eq. 4 as g1,2(2)= -1, g3,2(2)= 0, g5,2(2)= 1, g1,4(2)= 1, g3,4(2)= 0, and
[0183]
[0184] (2) (2) (2) (2) (2) g5,6(2)= — 1. The maximum gradients are g5,2(2)and g1,4(2). Since g5,2(2)and g1,4(2)are equal. Thus, randomly the pair of qubits (1,4) may be chosen. Thus, the state at stepr = 2 is given by:
[0185] = e‘4Z4riEq. 22 1 = — [|H - 1- + +) + | - 1 - F) - |+ + H - 1-) — |— + + + +)].
[0186]
[0187] In the fifth embodiment, the mapping of the active and inactive qubits may thus be updated to [1,2,4] and [3,5], respectively. The third set of gradients may be obtained, (3)n(3) Q (3) n (3)Q(3) using Eq. 4 as g3,1(3)= 0, g5,1(3)= −2, g3,2(3)= 0, g5,2(3)= 2, g3,4(3)= 0, and(3) (3)
[0188] g5,4(3)= —2. The maximum gradient at r = 2 correspond to g5,2(3)= 2. Thus, the state at step r = 3 is given by:
[0189] = e‘4Z2r5|<p2)
[0190] = -i=[|+- + + +) + |+ + + + -) + 1 -- + -+) + 1- + + - -)Eci 2\223
[0191]
[0192] - |+ + + - +) - |+ - +
[0193] -!-- + + -)]■
[0194] In the fifth embodiment, the mapping of the active and inactive qubits may thus be updated to [1,2,4,5] and [3], respectively. The fourth set of gradients may be obtained using Eq. 4 as g1,3(4)= 1, g2,3(4)= —1, g4,3(4)= 1, and g5,3(4)= — 1.
[0195] (4) (4)
[0196] The two maximum gradients are g1,3(4)and g4,3(4). The pair of qubits (3,4) is arbitrarily selected, which results in the state at r = 4:
[0197] |φ4〉 = ei(π / 4)Z₄Y₃|φ3〉
[0198] 1 = — [|H - 1- + +) + H - F)|+ + + H ) + |+ H - ) + | - 1 —
[0199]
[0200] 4Eq. 24
[0201] +> + l - + +) + 1- + + --) + 1- + - + -) - 1+ + + - +) - 1+ + - + +) - 1+ - + - -)
[0202] - 1+ -- + -)- 1- + + + +)- 1- + - - +)
[0203] - I-- + + -)-! - )]■
[0204] |<p4) may be written in the computational basis using its stabilizers being:
[0205] —XXXXX, -ZIIIIZ, +IZIIZ, -IIZIZ, -IIIZZ, which correspond to the resulting state in the computational basis:1 |<p4) = -p [| 10110> - 101001)]. Eq. 25
[0206]
[0207] N 2
[0208] Figure 3 illustrates an example setup of a MIMO transmitter 100 and a MIMO receiver 120. The Figure 3 illustrates an example of the MIMO transmitter 100, the quantum precoder 130, and the MIMO receiver 120, as illustrated in Figure 1 and described in the text relating thereto.
[0209] Using the quantum precoder 130, a perturbation vector v may be obtained as described in relation to method 200 of any of the first to fourth embodiments, based on the QUBO representation, the at least one user data symbol u, and the precoder matrix. The MIMO transmitter 100 may map the at least one user data symbol u to the signal constellation based on the modulation scheme to obtain at least one modulated symbol vectors. The MIMO transmitter 100 applies the perturbation vector v to reduce the transmission power. The at least one perturbed symbol vector s' may be obtained with:
[0210] s' = s + v.
[0211] The MIMO transmitter 100 may apply the precoder matrix P to the at least one perturbed symbol vector s' to obtain at least one precoder symbol b. The at least one precoder symbol b may be obtained with:
[0212] b = Ps’. - 27
[0213] In an example the at least one precoder symbol is assigned to an Orthogonal Frequency Division Multiplexing (ODFM) subcarrier to obtain a frequency-domain symbol vector. The frequency-domain vector is provided to the at least one Inverse Discrete Fourier Transform (IDFT) modulator 310A to 310C as shown in Figure 3. In an example, the at least one IDFT modulator 310A to 310C may apply the IDTF operation to convert the frequency-domain symbol vector to a time domain ODFM signal which then is prepared for transmission by passing through components to theantennas 1 to M for transmission to the MIMO receiver 120 via the communication path 140. After preparation for transmission, the transmitting 270 of the user data symbol may be performed. The components are illustrated as 320A to 320C in Figure 2 and may be at least one of at least one cyclic prefix addition unit, at least one digital-to-analog converter (DAC), at least one radio frequency (RF) up-conversion stage, and at least one power amplifier (PA). The MIMO receiver 120 may comprise the N antennas 340A to 340C. The IDFT modulator 310A to 310C, and the components 320A to 320C are optionally comprised in the MIMO transmitter 100 or alternatively are provided separately to the MIMO transmitter 100. In an example, the precoding vector may be determined in a periodic scheme corresponding to a scheduler. The scheduler may be a Medium Access Control (MAC) scheduler.
[0214] The MIMO transmitter 100 may be a device comprising antennas sending multiple data streams at the same time over the same frequency channels. The MIMO transmitter 100 may provide means for processing user data symbols using techniques like precoding, spatial multiplexing and beamforming.
[0215] The MIMO receiver 120 may be a device receiving multiple signals simultaneously using antennas comprised in the MIMO receiver 120. The MIMO receiver 120 may provide means for processing and decoding the received signals. The MIMO receiver 120 is a wireless communication device.
[0216] The MIMO system may comprise the MIMO transmitter 100 and the MIMO receiver 120 with each multiple transmit and receive antennas at both ends of the communication path 140. The MIMO system may enhance data throughput, signal quality, and / or spectral efficiency by, e.g., enabling the MIMO transmitter 100 to send multiple data streams simultaneously, leveraging multipath propagation, and / or utilizing spatial multiplexing.
[0217] The precoder matrix may be a mathematical transformation representing the characteristics of the radio channels between the MIMO transmitter 100 and the MIMO receiver 120.
[0218] The communication paths 140 may represent the radio channels between the MIMO transmitter 100 and the MIMO receiver 120. The radio channels may represent howsignals may propagate through an environment between the MIMO transmitter 100 and the MIMO receiver 120.
[0219] The user data symbol may correspond to data to be transmitted. The data is, for example, framed and scheduled in a MAC layer and mapped onto the modulation scheme used. At least one symbol in the modulation scheme may map to the at least one user data symbol.
[0220] The QUBO representation represents a mathematical optimization model used to solve combinational problems, e.g., finding the perturbation vector with lowest transmission power, by representing them as binary variables with quadratic interactions.
[0221] In an example, the quantum precoder 130 is the classical computer with access to a classical solver. The classical solver is a computational method which may be executed on the classical computer. The quantum precoder may comprise of a plurality of qubits and at least one ancilla qubit manipulated by Clifford gates to evolve into stabilizer states, which can be simulated efficiently classically. The quantum precoder may be a classically simulable quantum precoder. Alternatively, the quantum precoder 130 may be a quantum processor, such as a quantum processing unit (QPU) being capable of preparing or initializing qubits into quantum states using e.g. superposition or entanglement, applying quantum gates to process and measures the qubits collapsing the quantum state into a classical result. For example, the classical computer comprises a Central Processing Unit (CPU) and / or a Graphic Processing Unit GPU.
[0222] A qubit of the plurality of qubits or of the at least one ancilla qubit may be represented by a quantum-mechanical system such as a superconducting circuit, trapped ions, photons, atoms, nitrogen-vacancy centre diamonds, or quantum dots. The superconducting circuit may be used for qubit representation at low temperatures such as millikelvin. When using trapped ions, at least one ion may be trapped with an electromagnetic field to represent a qubit. In case of using photons, an integrated silicon or lithium niobate substrate may be used for preparing the photons. In case of using atoms, the atoms may be manipulated in an optical lattice for representation of the qubits. In case of using nitrogen-vacancy centre diamonds,a qubit representation may be exploited using defects in diamond lattices. In case of using quantum dots, semiconductor nanostructures may be used to confine electrons for the qubits representation. The qubit may exist in a quantum superposition of states 0 and 1, using the property of the quantum system, such as a polarization degree of freedom or a spin. The qubit may be represented by more than one quantum systems or a higher dimensional quantum system. Alternatively, or additionally, the qubit represents binary variables representing quantum information using the classical solver. The plurality of qubits representing the decision variables may represent non-zero diagonal entries in the QUBO matrix and / or field terms in the LFIM representation.
[0223] The at least one ancilla qubit, also referred to as auxiliary or extra qubit, is used to assist with an operation or task but is disregarded in a final outcome of a computation. The at least one ancilla qubit represents at least one auxiliary decision variable and is coupled with the single-spin term of LFIM model. The external field strength of the ancilla qubit is reused as the coupling strength in the enlarged ZFIM.
[0224] The perturbation vector is a vector representing changes and / or disturbances between the MIMO transmitter and MIMO receiver taking, e.g., noise and interference in the radio channels, into account.
[0225] The configuration of the plurality of qubits and the at least one ancilla qubit may may be a represented of the product state of the plurality of qubits and the at least one ancilla qubit at step r being a Quantum description. The product state may represent a quantum mechanical description representing information of positions of the plurality of qubits and the at least one ancilla qubit in superpositions of two basic states |0) and |1).
[0226] The final configuration of the plurality of qubits is the resulting states of the plurality of qubits which is determined after application of the ADAPT-Clifford algorithm.
[0227] The modulation order is a number of distinct user data symbols in the modulation scheme, such as BPSK, QPSK, 16-QAM, 64-QAM, 256-QAM, or 1024-QAM. More specifically, in the BPSK modulation scheme the modulation order is 2, in the QPSK the modulation scheme the modulation order is 4, etc.The complex elements of the perturbation vector represent individual complex numbers of the vector.
[0228] An Ising Model representation may represent a mathematical model of ferromagnetism in statistical mechanics. The LFIM representation is a variation of the Ising model. The LFIM representation contains an additional component to which a longitudinal external magnetic field is applied to the plurality of qubits. A ZFIM representation is another variation of the Ising model representation. In the ZFIM representation wherein no external magnetic field is applied to the plurality of qubits. The enlarged ZFIM representation may correspond to the ZFIM representation wherein no external field is applied to the plurality of qubits and the at least one ancilla qubit.
[0229] A stabilizer state such as the initial state, the product state, and the resulting state, may comprise states of the plurality of qubits and the at least one ancilla qubit. The stabilizer state may correspond to a joint eigenstate of a set of commuting Pauli gates, i.e., the Pauli gates in X, Y, Z direction. Thus, the stabilizer state may remain a stabilizer state under Clifford gates.
[0230] Figure 4 shows an example of a communication system 400.
[0231] In the example, the communication system 400 includes the communication network 402 that includes an access network 404, such as a radio access network (RAN), and a core network 406. The access network 404 includes one or more access network nodes or base stations of various types, e.g., base stations integrated in satellites. Some embodiments of the access network 404 may include more than one access network technology. The network node 410 of access network 404 facilitate direct or indirect connection of the wireless communication device, also referred to as user equipment (UE) 412 to the core network 406 over one or more wireless connections. In the example of a downlink scenario, the MIMO transmitter 100 may be comprised in the network node 410, and the MIMO receiver 120 in the UE 412. In an example of an uplink scenario, the MIMO transmitter 100 may be comprised in the UE 412, and the MIMO receiver 120 may be comprised in the network node 410. The network node 410 facilitates direct or indirect connection of one or more UEs 412 to the core network 406 over one or more wireless connections such as thecommunication path 140. Example wireless communications over a wireless connection include transmitting and / or receiving wireless signals using electromagnetic waves such as radio waves, infrared waves, light waves and / or other types of signals suitable for conveying information without the use of wires, cables, or other material conductors. Moreover, in different embodiments, the communication system 400 may include any number of wireless networks, network nodes, UEs, and / or any other components or systems that may facilitate or participate in the communication of data and / or signals whether via wireless connections. The communication system 400 may include and / or interface with any type of communication, telecommunication, data, cellular, radio network, and / or other similar type of system.
[0232] The UEs 412 may be any of a wide variety of communication devices, including the wireless communication device, arranged, configured, and / or operable to communicate wirelessly with the network node 410 and other communication devices. Similarly, the network node 410 are arranged, capable, configured, and / or operable to communicate directly or indirectly (e.g., via other devices of the communication network 402) with the UE 412 and / or with other network nodes or equipment in the communication network 402 to enable and / or provide network access, such as wireless network access, and / or to perform other functions, such as administration in the communication network 402. More specifically, UEs 412 may send messages, data, and / or other signals to network node 410 or other elements of the communication network 402 by transmitting such signals to the relevant device directly without the signals passing through any intervening devices or by transmitting such signals to the relevant device indirectly through an intervening device (or multiple intervening devices) that then transmit the signal to the relevant device. Similarly, the network node 410 may send messages, data, and other signals to UE 412, other network node 410, and other devices in the communication network 402 directly or indirectly.
[0233] As a whole, the communication system 400 of Figure 4 enables connectivity between the UE, network nodes, and hosts. In that sense, the communication system 400 may be configured to operate according to predefined rules or procedures, such as specific standards that include, but are not limited to thecommunication network such as Global System for Mobile Communications (GSM); Universal Mobile Telecommunications System (UMTS); Long Term Evolution (LTE), and / or other suitable 2G, 3G, 4G, 5G standards, or any applicable future generation standard (e.g., 6G); wireless local area network (WLAN) standards, such as the Institute of Electrical and Electronics Engineers (IEEE) 802.11 standards (Wi-Fi); and / or any other appropriate wireless communication standard, such as the Worldwide Interoperability for Microwave Access (Wi-Max), Bluetooth, Z-Wave, Near Field Communication (NFC) ZigBee, Li-Fi, and / or any low-power wide-area network (LPWAN) standards such as LoRa and Sigfox. Moreover, the communication system 400 may be configured to support multiple different standards, protocols, or other rule sets, with individual components supporting all of the relevant rule sets or with different components or sub-systems within the communication system 400 supporting different standards, protocols, or rule sets.
[0234] Figure 5 is a graph illustrating fractional change of a transmission power difference as a function of the number of antennas of the MIMO transmitter 100 (herein referred to as M). The transmission power may be determined by the MIMO transmitter 100 via the quantum precoder 130 by measuring the resulting state in step 250 of method 200. The transmission power herein is denoted with ^with ancilla- The fractional change of the transmission power difference is between the transmission power ^with anciiiaand a transmission power determined without a use of at least one ancilla qubit (herein referred to as ^without ancilla)ar|d is defined as:
[0235] ^without ancilla—^with ancilla 2Q
[0236]
[0237] ^without ancilla
[0238] The diagram of Figure 5 shows that the transmission power required for transmitting the user data symbol is reduced by using method 200, in comparison to a method without usage of at least one ancilla qubit. More particularly, Figure 5 shows that for a number of M = 8 antennas, the transmission power may be decreased about 70% by using the ancilla qubit compared to not using the ancilla qubit. Further, Figure 5 shows that for a number of M = 48 antennas, the fractional change may decrease in comparison toM = 8, wherein the fractional change saturates at around 20%reduction in the transmission power for higher antenna numbers. Thus, the transmission power required for transmission is reduced when using method 200 in comparison to a method without usage of at least one ancilla qubit.
[0239] Figure 6 illustrates two graphs illustrating a bit error rate (BER) as a function of a signal-to-noise ratio in decibel (dB) for different numbers of antennas of the MIMO transmitter 100 (herein referred to as M) using the BPSK modulation scheme and a channel matrix, wherein values in the channel matrix are simulated with gaussian noise. The signal-to-noise ratio per bandwidth ratio is herein referred to as Eb / N0, wherein Ebis an amount of energy that is used for transmission of a bit and Nois a noise power per unit bandwidth describing how much noise is present in a channel in the communication path. The BER may be obtained as described in Abdelbari, Amr, and Bulent Bilgehan, " The Derivation of The Probability of Error for BPSK, 16-QAM and 64-QAM in Rayleigh Fading Channel: A Unified Approach", arXiv:2406.16548, 2024. In an example of using a BPSK channel and assuming an Additive White Gaussian Noise (AWGN) channel, the BER may be obtained with:
[0240] Eq. 29
[0241] where erfc(x) = -^=f°° exdx.
[0242]
[0243] The left of the two graphs of Figure 6 shows an outcome using an embodiment of method 200, solving the vector precoding problem (VPP) using the Adapt-Clifford algorithm. A right graph of the two graphs of Figure 6 shows a solution using a Zero Forcing (ZF) method as described in Amor, Donia Ben, Michael Joham, and Wolfgang Utschick, 'Asymptotic behavior of zero-forcing precoding based on imperfect channel knowledge for massive miso fdd systems", ICC 2023-IEEE International Conference on Communications, 2500, IEEE, 2023. The two graphs show that the solution using the embodiment of method 200 provides a decreased BER over a solution using ZF.Figure 7 illustrates a block diagram illustrating embodiments of the MIMO transmitter 100 in further detail. The MIMO transmitter may comprise processing circuitry 704 and a computer readable storage medium 702. In practice, the method 200 performed by MIMO transmitter 100 are performed by processing circuitry 704, embodied in one or more processors and / or microprocessors arranged to execute a computer program 701 that is stored in a computer program product 705, here in the form of a computer readable medium 702. The computer readable medium 702 may be a memory, such as a flash memory, or a tangible non-volatile computer readable storage medium, such a Read-Only Memory (ROM) or a hard disk drive, or any combination thereof. The computer program 701 comprises computer-executable instructions stored in and downloaded to the computer readable medium 702 and are executable by the processing circuitry 704. Alternatively, the computer program 701 may be transferred to the computer readable medium 702 using a suitable computer program product, such as a memory stick or in a memory of a device.
[0244] Thus, the computer program 701 may be stored in any suitable manner in the computer program product. The processing circuitry 704 is arranged to cause the MIMO transmitter 100 to carry out the method 200 in accordance with any of the of the described embodiments. The processing circuitry 704 is in one embodiment one or more general-purpose processors wherein each one of the general purpose processors includes one or more cores, but may alternatively be a DSP, an ASIC, an FPGA, a CPLD, etc. An I / O interface 703 is provided for communicating with external and / or internal entities using wired communications, e.g., based on Ethernet, and / or wireless communications, e.g., Wi-Fi, and / or a cellular network corresponding to one or a combination of 5G cellular networks, LTE, LTE-advanced, UMTS, or any other current or future wireless network, such as a future 3GPP 6G network, as long as the principles described below are applicable.
[0245] While the MIMO transmitter 100 has been described in relation to Fig. 8 as being configured with a computer program 701, the MIMO transmitter 100 may alternatively be implemented in pure hardware, e.g., using one or more ASICs.
Claims
CLAIMS1. A method (200) performed by a Multiple-Input Multiple-Output, MIMO, transmitter (100), the method comprising:acquiring (210) at least one user data symbol to be transmitted to a MIMO receiver (120);acquiring (220) a precoder matrix representing characteristics of radio channels between the MIMO transmitter (100) and the MIMO receiver (120);determining (230), based on the at least one user data symbol and the precoder matrix, a Quantum Unconstrained Binary Optimization, QUBO, representation for a transmission power required for transmitting the user data symbol;embedding (240) the QUBO representation on a quantum precoder (130) comprising a plurality of qubits and at least one ancilla qubit;processing (250) the embedded QUBO representation using an ADAPT-Clifford algorithm until a configuration of the plurality of qubits and the at least one ancilla qubit is reached which minimizes the transmission power required for transmitting the user data symbol;determining (260) a perturbation vector based on a final configuration of the plurality of qubits; andtransmitting (270), to the MIMO receiver (120), the at least one user data symbol using the precoder matrix and the perturbation vector.
2. The method (200) according to claim 1, wherein a number of the plurality of qubits depends on a modulation order used for transmitting the at least one user data symbol.
3. The method (200) according any one of claims 1 or 2, wherein the determining (260) the perturbation vector comprises:acquiring (262) the final configuration of the plurality of qubits from the quantum precoder (130); anddetermining (264) complex elements of the perturbation vector based on the final configuration of the plurality of qubits.
4. The method (200) according to any one of claims 1 to 3, wherein the embedding (240) the QUBO representation comprises:determining (242) a Longitudinal Field Ising Model, LFIM, representation based on the QUBO representation; andembedding (244) the LFIM representation in an enlarged Zero Field Ising Model, ZFIM, representation,wherein the ZFIM representation is embedded on the quantum precoder (130).
5. The method (200) according to claim 4, wherein the processing (250) the embedded QUBO representation using the ADAPT-Clifford algorithm comprises:applying (252) the Adapt-Clifford algorithm on the enlarged ZFIM representation.
6. The method (200) according to any of claims 1 to 5, wherein the embedding (240) the QUBO representation comprises preparing (246) the plurality of qubits and at least one ancilla qubit based on the QUBO representation.
7. The method (200) according to any of claims 1 to 6, wherein the processing (250) the embedded QUBO representation using the ADAPT- Clifford algorithm comprisesapplying (254) a Clifford Gate to each of the plurality of qubits and the at least one ancilla qubit based on the QUBO representation.
8. The method (200) according to any of claim 1 to 7, wherein the quantum precoder (130) is a classical computer with access to a classical solver, wherein the configuration of the plurality of qubits and the at least one ancilla qubit correspond to a plurality of decision variables, and wherein processing the QUBO representation corresponds to determining the decision variables.
9. A Multiple-Input Multiple-Output, MIMO, transmitter (100) configured to:acquire at least one user data symbol to be transmitted to a MIMO receiver (120);acquire a precoder matrix representing characteristics of radio channels between the MIMO transmitter (100) and the MIMO receiver (120);determine, based on the at least one user data symbol and the precoder matrix, a Quantum Unconstrained Binary Optimization, QUBO, representation for a transmission power required for transmitting the user data symbol;embed the QUBO representation on a quantum precoder (130) comprising a plurality of qubits and at least one ancilla qubit;process the embedded QUBO representation using an ADAPT-Clifford algorithm until a configuration of the plurality of qubits and the at least one ancilla qubit is reached which minimizes the transmission power required for transmitting the user data symbol;determine a perturbation vector based on a final configuration of the plurality of qubits; andtransmit, to the MIMO receiver (120), the at least one user data symbol using the precoder matrix and the perturbation vector.
10. A computer program (701) comprising instructions which, when executed on processing circuitry (704) of a Multiple-Input Multiple-Output, MIMO, transmitter (100), cause the processing circuitry (704) of the MIMO transmitter (100) to carry out the method according to any one of claims 1 to 8.
11. A computer-readable medium (702) comprising instructions that, when executed by processing circuitry (704) of a Multiple-Input Multiple-Output, MIMO, transmitter (100), cause the processing circuitry (704) of the MIMO transmitter (100) to carry out the method according to any one of claims 1 to 8.